A Variational Analysis of Some Classes of [615006]

A Variational Analysis of Some Classes of
Integral and Differential Equations:
Eigenvalue Problems and Torsional Creep
Problems
PhD. Candidate: Maria F arc a seanu
Submitted to
Department of Mathematics
University of Craiova
In partial fullment of the requirements for the degree of
Doctor of Philosophy in Mathematics
Supervisor: Mihai Mih ailescu
Craiova, Romania
2018

i
Acknowledgements
First and foremost, I would like to express my sincere gratitude to my supervisor, Professor Mihai
Mih ailescu, for accepting me as Ph.D. student: [anonimizat], I would like to oer my special thanks to Dr. Denisa Stancu-Dumitru for her collaboration
and valuable advices during my Ph.D. thesis. I am deeply indebted to her.
I am grateful to the members of my "comisie de indrumare" for their professional support and
guiding in my research.
This work would not have been possible without the support of the following nancial sources:
Doctoral Fellowship of the Doctoral School of Sciences from the University of Craiova (October
2015 – October 2018);
The research project: Variable Exponent Analysis: Partial Dierential Equations and Calculus
of Variations (founded by CNCS-UEFISCDI; project number PN-II-ID-PCE-2012-4-0021; host insti-
tution: IMAR; project leader: Mihai Mih ailescu; October 2015- September 2016);
The research project: Analysis of Schr odinger Equations (founded by CNCS-UEFISCDI; project
number PN-II-RU-TE-2014-4-0007; host institution: IMAR; project leader: Ioan-Liviu Ignat; October
2015 – October 2017);
The research project: Typical and Nontypical Eigenvalue Problems for Some Classes of Dier-
ential Operators (founded by: CNCS-UEFISCDI; project number PN-III-P4-ID-PCE-2016-0035; host
institution: IMAR; project leader: Mihai Mih ailescu; July 2017 – December 2019).
I would also like to thank to the members of the above research projects for their many valuable,
helpful and interesting discussions also beyond mathematics.
I am indebted to my professor from High School, Daniel Ion, for encouraging me to continue with
the Ph.D. studies.
Special thanks to Dr. Dana Mih ailescu, for reading this thesis and helping me in improving the
language style of it.
I am also grateful to all my good friends who always believe in me and cheer me up.
Last, but not least, I want to express my appreciation and gratitude to my mother for her uncon-
ditional love, patience and support.

Contents
1 Introduction 1
1.1 Motivation and thesis aims . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Elements of Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 The Direct Method in the Calculus of Variations . . . . . . . . . . . . . . . . . . 4
1.2.2 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Eigenvalue problems 9
2.1 An eigenvalue problem involving an anisotropic dierential operator . . . . . . . . . . . 9
2.2 Fractional eigenvalue problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2.1 On an eigenvalue problem involving the fractional ( s;p)-Laplacian . . . . . . . . 17
2.2.2 Perturbed fractional eigenvalue problems . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Continuity of the rst eigenvalue for a family of degenerate eigenvalue problems . . . . . 34
2.4 A maximum principle for a class of rst order dierential operators . . . . . . . . . . . . 39
3 Torsional creep problems 47
3.1 On the convergence of the sequence of solutions for a family of eigenvalue problems . . . 48
3.2 On a family of torsional creep problems involving rapidly growing operators in divergence
form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.1 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2.2 Variational solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.3 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.4 A -convergence result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.2.5 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.3 On a family of torsional creep problems in Finsler metrics . . . . . . . . . . . . . . . . . 69
3.3.1 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.3.2 Finsler norms: denition, properties, examples . . . . . . . . . . . . . . . . . . . 70
3.3.3 Proof of Theorem 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.3.4 Proof of Theorem 3.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
ii

iii
4 Final comments and further directions of research 84
4.1 \Can one hear the shape of a drum?" . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.2 \Hot spots conjecture" . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.3 The spectrum of the p-Laplace operator when p6= 2 . . . . . . . . . . . . . . . . . . . . 85
4.4 The uniqueness of the limit of the sequence of principal eigenfunctions of p-Laplacian as
p!1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Chapter 1
Introduction
1.1 Motivation and thesis aims
A partial dierential equation (PDE) is an equation involving an unknown function of two or more
variables and certain of its partial derivatives. PDE's appear frequently in all domains, such as physics,
mechanics and engineering. In fact, whenever we have an interaction between some independent vari-
ables, we attempt to nd functions using these variables and to shape a multitude of processes by
developing equations for these functions. Consequently, due to the rich variety of phenomena which
can be modeled by PDE's, there is no general theory known concerning the solvability of all of them.
There are many methods to solve PDE's, each method being applicable to a certain class of equa-
tions. Solving a given PDE depends in large part on the particular structure of the problem at hand. It
is considered that a given problem is well-posed if it has a solution which is unique and stable (i.e. the
solution depends continuously on the data given in the problem). There are many dierent denitions of
the solution for a PDE. The most natural notion of solution arises when all the derivatives which appear
in the statement of the PDE exist and are continuous, although maybe certain higher derivatives do
not exist. This kind of solutions are called \ classical " solutions. On the other hand, there are functions
for which the derivatives may not all exist, but which satisfy the equation in some precisely dened
sense. These functions are known in the literature as \ weak" solutions and they are most often used
in the analysis of PDE's. However, even in situations where an equation has dierentiable solutions,
it is often convenient to prove rst the existence of weak solutions and only later to show that those
solutions are in fact smooth enough.
In general, the (weak) solutions can be found as critical points of the corresponding variational
functionals dened on an appropriate function space dictated by the data of the problem. The simplest
way to obtain such a critical point, is to look for a global extremum, which in most of the cases is a
global minimum. If the functional has good properties, such as the smoothness or the boundedness,
the existence of the minimum points can be obtained by applying direct methods in the calculus of
variations. Otherwise, for example, the lack of smoothness can be tackled by a reformulation of the
problem as a variational inequality, or if the functional is unbounded, there exist some minimization
1

1. Introduction 2
techniques that can still be protably used, by constraining the functional on a set where it is bounded
from below. Typical examples of such techniques are minimization on Spheres, or on the Nehari
manifold.
Some of the fundamental problems in mathematical physics are, probably, the eigenvalue problems
for elliptic PDE's. The analysis of such equations involves, in general, energy methods which are based
on the critical point theory that has been mentioned previously. For example, the eigenvalue problem
for thepLaplace operator subject to zero Dirichlet boundary condition, i.e.
8
><
>:
pu=jujp2uin
u= 0 on @
;(1.1)
where
is a bounded domain in RN,p2(1;1) and
pu:= div(jrujp2ru) stands for the pLaplacian,
has been studied extensively along the time and many interesting results have been obtained. If p= 2,
problem (1.1) becomes the eigenvalue problem for the Laplacian, that is
8
><
>:
u=u in
u= 0 on @
;
and it is well-known that all the eigenvalues are positive and form an increasing and unbounded sequence
0<  1<  23:::such thatn!+1asn!1 . Moreover, in this particular case, all the
eigenvalues have nite multiplicities and the rst one is simple. For p6= 2 andN2, the complete
description of the set of all eigenvalues is an open problem. It is known that the Ljusternik-Schnirelman
theory ensures the existence of an innite sequence of positive eigenvalues of problem (1.1), but in
general this theory does not provide all eigenvalues. However, it can be shown, the existence of a
principal eigenvalue, 1(p), that is the smallest of all possible eigenvalues , which can be characterized
from a variational point of view in the following manner
1(p) := inf
u2C1
0(
)nf0gR

jrujpdxR

jujpdx:
Moreover,1(p) is simple, isolated and the corresponding eigenfunctions are minimizers of 1(p), that
do not change sign in
. Also, it was showed that, if up>0 is a solution of (1.1), then there exists a
subsequence offupg, which converges uniformly in
, when p!1 , to a nontrivial and nonnegative
solution, dened in the viscosity sense, of the limiting problem
8
><
>:minfjruj1u;
1ug= 0 in
u= 0 on @
;
where
1u:=hD2uru;ruistands for the1Laplace operator and
1:=1
max
x2
dist(x;@
):

1. Introduction 3
Motivated by the above results that have been obtained in the case of the pLaplace operator, the
rst part of this thesis (Chapter 2) is devoted to the study of various eigenvalue problems which involve
dierent types of elliptic partial dierential operators or integral operators. For instance, we consider
an anisotropic version of the pLaplacian, that is the ( p;q)Laplace operator, dened by

p;qu:= divx
jrxujp2rxu
+ divy
jryujq2ryu
;
where we have denoted by rxuandryuthe derivatives of uwith respect to the rst Lvariables and
with respect to the last Mvariables (L+M=N) and a fractional version of the pLaplacian, called
fractional (s;p)Laplacian, given by
(
p)su(x) := 2 lim
&0Z
jxyjju(x)u(y)jp2(u(x)u(y))
jxyjN+spdy; x2RN;
where 1< p <1and 0< s < 1. To each of these operators, we associate adequate eigenvalue
problems and we characterize their spectrum using methods based on critical point theory. Besides
that, in this chapter, we study the continuity of the rst eigenvalue with respect to a parameter, for
a family of degenerate eigenvalue problems and, in the end, we give a maximum principle for a class
of rst order dierential operators, using as starting point an eigenvalue problem for elliptic operators
involving variable exponent growth conditions.
The second part of this thesis (Chapter 3) is devoted to the study of some PDE's that are connected
with the concept of \torsional creep". This phenomenon is explained as being the permanent plastic
deformation of a material subject to a torsional moment for an extended period of time and at suciently
high temperature. The modelling of such a phenomenon is related to inhomogeneous problems of the
type 8
><
>:
pu= 1 in
u= 0 on @
;(1.2)
whenp!1 . It is known that problem (1.2) possesses a unique solution, up, which uniformly converges
to function dist(
;@
) (that is the distance function to the boundary of
), as p!1 . Note that,
the limit case is of special interest in applications, since it models the perfect plastic torsion. In this
chapter, our aim will be the study of the asymptotic behaviour of some families of solutions for dierent
equations, which represent extensions of the classical torsional creep problem (1.2).
1.2 Elements of Calculus of Variations
In this section we collect without proof some basic results in the Calculus of Variations which will be
essential in our subsequent analysis.

1. Introduction 4
1.2.1 The Direct Method in the Calculus of Variations
The following theorem plays an important role in the analysis of the existence of solutions for dierent
classes of equations.
Theorem 1.1. (see [99, Theorem 2.1]) Suppose Vis a re
exive Banach space with norm k
k and let
MVbe a weakly closed subset of V. SupposeE:M!R[f+1gis coercive on Mwith respect to
V, that is
(1)E(u)!1 askuk!1 ,u2M
and (sequentially) weakly lower semi-continuous on Mwith respect to V, that is
(2) for any u2M, any sequence (um)inMsuch thatumconverges weakly to uinVthere holds
E(u)lim
m!1E(um):
ThenEis bounded from below on Mand attains its inmum in M
Remark. IfVis the dual of a separable normed vector space, Theorem 1.1 remains valid if we replace
weak byweak?-convergence.
1.2.2 Convergence
This section contains some denitions and results regarding the concept of convergence that are
useful throughout this thesis. The notion of convergence, which was introduced by Ennio De Giorgi
in the 1970s, has become the most
exible and natural notion of converge for variational problems.
The convergence of the corresponding functional for some elliptic PDE's is a tool to obtain a limit
functional and, also, the convergence of critical points of the functional to a critical point of the limit
functional. More details regarding this topic can be found, for instance, in [31] and [22]. We start this
section by recalling the denition of the concept of -convergence (introduced in [33], [34]) in metric
spaces.
Denition 1.1. LetXbe a metric space. A sequence fFngof functionals Fn:X!R:=R[f+1g
is said to (X)-converge to F:X!R, and we write (X)lim
n!1Fn=F, if the following hold:
(i)for everyu2XandfungXsuch thatun!uinX, we have
F(u)lim inf
n!1Fn(un) ;
(ii) for everyu2X, there exists a recovery sequence fungX, such that un!uinXand
F(u)lim sup
n!1Fn(un):
The following results are well-known and can be found, for example, in [66, Lemma 6.1.1] and [66,
Corollary 6.1.1].

1. Introduction 5
Proposition 1.1. LetXbe a topological space that satises the rst axiom of countability, and assume
thatfungis a sequence, such that un!uinXasn!1 ,
lim sup
n!1F(un)F(u);
and such that, for every m2N, there exists a sequence fum;ngn;um;n!umasn!1 , with
lim sup
n!1Fn(um;n)F(um):
Then, there exists a recovering sequence for uin the sense of (ii) of Denition 1.1.
Proposition 1.2. LetXbe a topological space satisfying the rst axiom of countability, and assume
that the sequence fFngof functionals Fn:X!R,converges toF:X!R. Letznbe a minimizer
forFn. Ifzn!zinX, thenzis a minimizer of F, and
F(z) = lim inf
n!1Fn(zn):
1.3 Outline of the thesis
This thesis is structured into 3 chapters (Chapters 2-4). Chapters 2 and 3 represent the main body
of the thesis, presenting the main results of our research. Chapter 4 contains some open problems on
the topic of the thesis that represent the starting point for our further research. In the following we
describe in brief the main results from this thesis.
Throughout this thesis, we consider that
is a bounded domain from RNwith smooth boundary
@
.
Chapter 2 is devoted to the study of various eigenvalue problems, which involve dierent types
of dierential or integral operators. In this chapter, denotes a real parameter, which will be called
an eigenvalue of a problem if that problem has a nontrivial solution. This chapter contains 4 sections
(Sections 2.1-2.4).
Section 2.1 (based on paper [43]) is concerned with the study of an eigenvalue problem involving an
anisotropic ( p;q)-Laplacian. More precisely, if LandMare two positive integers, such that L+M=N,
then for each two real numbers pandq, satisfying 1 <p<q<1, and each smooth function u:
!R,
we dene the anisotropic (p;q)-Laplacian by

p;qu:= divx
jrxujp2rxu
+ divy
jryujq2ryu
;
where we have denoted by rxuandryuthe derivatives of uwith respect to the rst Lvariables and
with respect to the last Mvariables, respectively, that is,
rxu=@u
@x1;@u
@x2;:::;@u
@xL
andryu=@u
@y1;@u
@y2;:::;@u
@yM
:

1. Introduction 6
The goal of Section 2.1, is to study the existence of nontrivial solutions of the following anisotropic
eigenvalue problem 8
><
>:
p;qu=jujq2u;in
u= 0; on@
:
We show that the set of eigenvalues of the above problem is exactly an unbounded open interval (see
Theorem 2.1).
In Section 2.2 (based on papers [44] and [49]), we study two eigenvalue problems involving an
integral operator. This section is divided into two subsections: 2.2.1 and 2.2.2. In order to present
the main results from these subsections, we dene for each p2(1;1) ands2(0;1), the fractional
(s;p)-Laplace operator by
(
p)su(x) := 2 lim
&0Z
jxyjju(x)u(y)jp2(u(x)u(y))
jxyjN+spdy; x2RN:
In Subsection 2.2.1, we investigate the problem
8
><
>:(
p)su(x) =f(x;u(x));forx2
u(x) = 0; forx2RNn
;(1.3)
where function f:
R!Ris given by
f(x;t) =8
><
>:h(x;t);ift0;
jtjp2t;ift<0:
Functionh:
[0;1)!Ris a Caratheodory function, which satises some convenient hypotheses
(see conditions (H1)-(H3) from Subsection 2.2.1). We show that problem (1.3) possesses, on the one
hand, a continuous family of eigenvalues and, on the other hand, one more eigenvalue, which is isolated
in the set of eigenvalues of the problem (see Theorem 2.2).
Next, in Subsection 2.2.2, we study the following perturbed eigenvalue problem
8
><
>:(
p)su(x) + (
q)tu(x) =ju(x)jr2u(x);forx2
u(x) = 0; forx2RNn
;(1.4)
wheres,t,pandqare real numbers satisfying the assumption
0<t<s< 1;1<p<q<1; sN
p=tN
q;
andr2fp;qg. Our purpose is to determine all the parameters , for which problem (1.4) possesses

1. Introduction 7
nontrivial weak solutions. With that end in view, dene
1:=8
>>>>>><
>>>>>>:infu2C1
0(
)nf0g[u]p
Ws;p(RN)Z
RNjujpdx;ifr=p
infu2C1
0(
)nf0g[u]q
Wt;q(RN)Z
RNjujqdx;ifr=q;
where [
]Ws;p(RN)and [
]Wt;q(RN)stand for the Gagliardo seminorms in the fractional Sobolev spaces
Ws;p
0(RN) andWt;q
0(RN), respectively. In Theorem 2.3, we show that the set of all real parameters for
which problem (1.4) has at least a nontrivial weak solution is exactly the interval ( 1;1). Moreover,
the corresponding weak solutions could be chosen to be non-negative.
In Section 2.3 (based on paper [45]), for each 2[0;2), we consider the eigenvalue problem
8
><
>:div(jxjru) =u; forx2
u= 0; forx2@
;
where 02
. Denoting by 1() the rst eigenvalue of the problem, we prove the continuity of the
function1with respect to on the interval [0 ;2) (see Theorem 2.4).
The goal of Section 2.4 (based on paper [47]) is to present how a series of results obtained in
connection with an eigenvalue problem involving variable exponents can be used in order to obtain
a maximum principle, which complements the classical maximum principle for elliptic operators (see
Theorem 2.7).
Chapter 3 is divided into 3 sections and devoted to the study of some PDE's that are related with
the concept of torsional creep.
In Section 3.1 (based on paper [50]), we continue to keep the connection with the previous chapter,
by considering, for each integer n1, the family of eigenvalue problems
8
>>>><
>>>>:
2nu=u; forx2
u= 0; forx2@
jjujjL2(
)= 1;
where
2nu:= div(jruj2n2ru) is the 2n-Laplace operator and is a real number. For each integer
n2, we prove the existence of a lowest eigenvalue of the problem, and then, that the sequence of
corresponding eigenfunctions converges uniformly on
to kk1
L2(
), where(x) := infy2@
jxyj,
8×2
, denotes the distance function to the boundary of
(see Theorem 3.1).
The goal of Section 3.2 (based on paper [46]), is to investigate the asymptotic behaviour of the
family of solutions for the following family of equations
8
><
>:div
'n(jruj)
jrujru
='n(1) in
u= 0 on @
;(1.5)

1. Introduction 8
where, for each integer n>1, the mappings 'n:R!Rare odd, increasing homeomorphisms of class
C1dened by
'n(t) :=pnjtjpn2tejtjpn;8t2R;
wherepn2(1;1) are given real numbers, such that lim n!1pn= +1. We show that the sequence of
solutions of equations (1.5) converges uniformly to the distance function to the boundary of the domain
(see Theorem 3.3).
In Section 3.3 (based on paper [51]), we consider H:RN![0;1) a Finsler norm and :
R!
(0;1) a continuous function for which there exist two positive constants ;, such that
0<(x;t)<+1;8×2
;8t2R:
For each real number p2(N;1), we consider the following problem
8
><
>:div((x;u)H(ru)p2H(ru)) =f; x2
;
u= 0; x 2@
;(1.6)
wheref:
!(0;1) is a given continuous function and H:RN!RNis dened by
Hi() :=@
@i1
2H()2
;82RN;8i2f1;:::;Ng:
We show that the family of solutions of equations (1.6), converges uniformly to the distance function
to the boundary of the domain, which takes into account the dual of the Finsler norm involved in the
equation (see Theorems 3.5 and 3.6).
Chapter 4 presents some open problems related to the topic of this thesis which will guide our
further research.

Chapter 2
Eigenvalue problems
The goal of this chapter is to analyze some eigenvalue problems.
2.1 An eigenvalue problem involving an anisotropic dierential op-
erator
The prototype anisotropic Laplace operator has the following expression
NX
i=1@
@xi @u
@xipi2@u
@xi!
; (2.1)
wherep1,…,pN2(1;1) are real numbers. Such operators were largely studied in the literature mainly
due to the fact that they can model with sucient accuracy some phenomena which can occur in
dierent branches of science. For instance, according to V etois [105], if we consider the time evolution
version of the anisotropic Laplace operator, then it can serve as a model for the dynamics of
uids
in anisotropic media, when the conductivities of the media are dierent in distinct directions (see
Antontsev, D az & Shmar ev [5]), or it can appear in mathematical biology as a model for the propagation
of epidemic diseases in heterogeneous domains (see Bendahmane & Karlsen [13] and Bendahmane,
Langlais & Saad [14]).
The goal of this section is to study the existence of nontrivial solutions of the following anisotropic
eigenvalue problem 8
><
>:
p;qu=jujq2u;in
u= 0; on@
;(2.2)
where
p;q(
) represents the anisotropic (p;q)-Laplacian , which for each two real numbers pandq,
satisfying 1 <p<q<1, and each smooth function u:
!Ris dened by

p;qu:= divx
jrxujp2rxu
+ divy
jryujq2ryu
:
9

2. Eigenvalue problems 10
Here, we have denoted by rxuandryuthe derivatives of uwith respect to the rst Lvariables and
with respect to the last Mvariables, respectively, that is,
rxu=@u
@x1;@u
@x2;:::;@u
@xL
andryu=@u
@y1;@u
@y2;:::;@u
@yM
;
whereLandMare two positive integers such that L+M=N. Note that, the anisotropic ( p;q)-
Laplacian has similar properties with the operator given by relation (2.1) in the particular case when
p1=p2=:::=pL=pandpL+1=pL+2=:::=pN=q, that is
LX
i=1@
@xi @u
@xip2@u
@xi!
+NX
i=L+1@
@xi @u
@xiq2@u
@xi!
:
Many interesting equations involving the anisotropic ( p;q)-Laplacian were studied in recent years. In
this context we just remember the papers by Di Castro, P erez-Llanos & Urbano [39] or P erez-Llanos
& Rossi [94].
In the following, we recall a few known results in the eld which are directly related to equations
of type (2.2). Thus, if we replace the exponent qfrom the right-hand side of equation (2.2) by a real
numberrthen the problem 8
><
>:
p;qu=jujr2u;in
u= 0; on@
;(2.3)
withr > q was investigated by Fragala, Gazzola & Kawohl in [52]. The case when ris positive
and subcritical was studied by Tersenov & Tersenov in [102], while the case when p < r < q was
discussed by Di Castro & Montefusco in [38]. In all these papers, the case when in the right-hand
side of equation (2.3) appears the growth q, which is also involved in the left-hand side of the problem
was not considered. Thus, our study comes to complement the above mentioned results. In a close
context our study also complements the results by Mih ailescu, Pucci & R adulescu [88] and Mih ailescu,
Moro sanu & R adulescu [86], obtained on similar types of eigenvalue problems. Finally, we note that
in the case when the anisotropic ( p;q)-Laplacian from the left-hand side of equation (2.2) is replaced
by the (p;q)-Laplacian operator, that is
pu+
qu= div((jrujp2+jrujq2)ru), equations of type
(2.2) were studied in [84, 48, 18, 85, 91].
In order to investigate equation (2.2), we introduce an appropriate anisotropic Sobolev function
space. We will denote by W1;p;q
0(
), with 1 < p < q <1, the anisotropic Sobolev space obtained as
the completion of the space C1
0(
) with respect to the norm
kukW1;p;q
0(
):=krxukLp(
)+kryukLq(
):
Sincep<q , it is obvious that W1;p;q
0(
) is continuously embedded in the usual Sobolev space W1;p
0(
).
Actually, an equivalent denition for W1;p;q
0(
) could be
W1;p;q
0(
) :=fu2W1;p
0(
) :jrxujp;jryujq2L1(
)g:

2. Eigenvalue problems 11
By [52, Theorem 1] (see also [103]), we have that W1;p;q
0(
) is compactly embedded in Lq(
) provided
that the constants L,M,pandqinvolved in equation (2.2) satisfy
L
p+M
q>1 andL
pL
q<1: (2.4)
Note that, in the case when pNthe anisotropic Sobolev space W1;p;q
0(
) is compactly embedded in
Lq(
). Also, we recall a Poincar e-type inequality given in [52, relation (11) on p. 722], namely
kukLr(
)air
2

@u
@xi

Lr(
);8u2C1
0(
);8r>1;8i2f1;:::;Ng; (2.5)
whereai:= supx;y2
(xy;ei), is the width of
in direction ei(herefe1;:::;eNgstands for the canonical
basis in RN).
An important role in our analysis will be played by the following constant
1(q) := inf
u2W1;p;q
0(
)nf0gZ

jryujq
Z

jujq: (2.6)
By inequality (2.5), we deduce that
1(q)>0:
We say that 2Ris an eigenvalue of problem (2.2), if there exists u2W1;p;q
0(
)nf0g, such that
Z

jrxujp2rxurxv+Z

jryujq2ryuryv=Z

jujq2uv; (2.7)
for allv2W1;p;q
0(
).
The main result of this section is given by the following theorem.
Theorem 2.1. (see [43, Theorem 1]) Assume that 1< p < q <1and eitherpNor (2.4) holds.
Then the set of eigenvalues of problem (2.2) is given exactly by the open interval (1(q);1).
Proof of Theorem 2.1. The conclusion of Theorem 2.1 will follow as a consequence of two
propositions that will be presented below. In the rst one, we establish the nonexistence of eigenvalues in
the interval (1;1(q)), while in the second one, the existence of eigenvalues in the interval ( 1(q);1).
Proposition 2.1. Each2(1;1(q)]is not an eigenvalue of problem (2.2).
Proof. If0, the conclusion of the proposition is obvious by testing in relation (2.7) with v=u.
Assume by contradiction that there exists 2(0;1(q)) for which problem (2.2) possesses a solution
u2W1;p;q
0(
)nf0g. By relations (2.6) and (2.7) with v=u, we get
0<(1(q))Z

jujqZ

jryujqZ

jujq
Z

jrxujp+Z

jryujqZ

jujq
= 0;

2. Eigenvalue problems 12
which represents a contradiction. Hence, any 2(0;1(q)) cannot be an eigenvalue of problem (2.2).
To complete the proof of the proposition, we will show that 1(q) cannot be an eigenvalue of problem
(2.2). Indeed, if we assume again by contradiction that there exists u1(q)2W1;p;q
0(
)nf0gsuch that
(2.7) holds with =1(q), then letting v=u1(q)in (2.7) we have
Z

rxu1(q)p+Z

ryu1(q)q=1(q)Z

ju1(q)jq:
On the other hand, by (2.6) it is clear that
1(q)Z

ju1(q)jqZ

ryu1(q)q:
The last two relations yield Z

rxu1(q)p= 0;
which combined with the fact that u1(q)2W1;p;q
0(
),!W1;p
0(
),!Lp(
) (see also (2.5)), impliesR

ju1(q)jp= 0, and hence u1(q)= 0, a contradiction with u1(q)2W1;p;q
0(
)nf0g.
Proposition 2.2. Every2(1(q);1)is an eigenvalue of problem (2.2).
Proof. Fix> 1(q) and dene J:W1;p;q
0(
)!Rby
J(u) :=1
pZ

jrxujp+1
qZ

jryujq
qZ

jujq:
Standard arguments can be used in order to deduce that J2C1(W1;p;q
0(
);R) with the derivative
given by
hJ0
(u);vi=Z

jrxujp2rxurxv+Z

jryujq2ryuryvZ

jujq2uv;
for allu;v2W1;p;q
0(
). We note that is an eigenvalue of problem (2.2), if and only if Jpossesses
a nontrivial critical point. In order to nd it, we will analyze the energy functional J, on a so-called
Nehari manifold (see, e.g., [8, Section 2.3.3] or [101] for a general description of the method). Thus,
consider the Nehari manifold
N:=fu2W1;p;q
0(
)nf0g:hJ0
(u);ui= 0g
=
u2W1;p;q
0(
)nf0g:Z

jrxujp+Z

jryujq=Z

jujq
:
Note that, onNthe functional Jhas the following expression
J(u) =1
pZ

jrxujp+1
qZ

jryujq
qZ

jujq
=1
p1
qZ

jrxujp:
Set
m:= inf
w2NJ(w):

2. Eigenvalue problems 13
Sincep<q we deduce that m0.
We begin by checking that working on the Nehari manifold makes sense, that is N6=;. Since
> 1(q), by the denition of 1(q), we deduce that there exists v2W1;p;q
0(
)nf0gfor which
Z

jryvjq<Z

jvjq:
Then, there exists t>0, such that tv2N, i.e.
tpZ

jrxvjp+tqZ

jryvjq=tqZ

jvjq;
which is obvious for
t=0
BB@Z

jvjqZ

jryvjq
Z

jrxvjp1
CCA1
pq
:
The above computations show that Nis not empty.
In the following, we establish some properties of the functional Jon the Nehari manifold.
Claim 1. Every minimizing sequence for mis bounded .
Indeed, letfungNbe a minimizing sequence for m, i.e.
0<Z

junjqZ

jryunjq=Z

jrxunjp!1
p1
q1
m;asn!1: (2.8)
Assume by contradiction thatR

jryunjq!1 asn!1 . Then, by (2.8), we deduce thatR

junjq!1
asn!1 , too. Setvn:=un
kunkLq(
). SinceR

jryunjq<R

junjq, we deduce thatR

jryvnjq<, for
eachn. Thus,fryvngnis bounded in Lq(
).
Dividing (2.8) by jjunjjp
Lq(
), we get
Z

jrxvnjp=Z

junjqZ

jryunjq
jjunjjp
Lq(
)!0 asn!1;
sinceR

junjqR

jryunjq!
1
p1
q1
mandkunkp
Lq(
)!1 . It follows thatfrxvngnis bounded
inLp(
). By the fact that fryvngnandfrxvngnare bounded in Lq(
), respectively, in Lp(
),
we deduce that there exists v02W1;p;q
0(
) such that, after eventually extracting a subsequence, vn
converges weakly to v0inW1;p;q
0(
) and strongly in Lp(
) andLq(
) (the strong convergence is a
consequence of relation (2.4) which assures the compact embedding of W1;p;q
0(
) inLp(
) andLq(
)).
By the weak lower semicontinuity of the W1;p;q
0 norm, we have
Z

jrxv0jplim inf
n!1Z

jrxvnjp= 0;

2. Eigenvalue problems 14
and which in view of inequality (2.5) implies v0= 0. In particular, it follows that vn!0 inLq(
),
but this is a contradiction with the fact that kvnkLq(
)= 1, for each n. Consequently,fungshould be
bounded in W1;p;q
0(
).
Claim 2.m:= infw2NJ(w)>0.
Assume by contradiction that m= 0. LetfungNbe a minimizing sequence for m. Then we
have
0<Z

junjqZ

jryunjq=Z

jrxunjp!0;asn!1: (2.9)
ByClaim 1 we deduce thatfungis bounded in W1;p;q
0(
). It follows that there exists u02W1;p;q
0(
)
such that, after eventually extracting a subsequence, un* u 0inW1;p;q
0(
) andun!u0inLq(
).
Thus, Z

jrxu0jplim inf
n!1Z

jrxunjp= 0:
That fact and inequality (2.5) imply u0= 0. Thus, we found that un*0 inW1;p;q
0(
) andun!0 in
Lq(
). Setvn:=unkunk1
Lq(
). By (2.9), we haveR

jryunjq<R

junjq, which yieldsR

jryvnjq<.
Relation (2.9) also implies that
Z

jrxunjpZ

junjq=kunkq
Lq(
);
which leads to Z

jrxvnjpkunkqp
Lq(
)!0;asn!1;
sincep<q . Thus,fvngis bounded in W1;p;q
0(
). It follows that there exists v02W1;p;q
0(
) such that,
after eventually extracting a subsequence, vn*v 0inW1;p;q
0(
) andvn!v0inLq(
). Dividing (2.9)
bykunkq
Lq(
), we nd
Z

jrxvnjp=kunkqp
Lq(
)
Z

jryvnjq
!0;asn!1:
In the above relation, we took into account that 0 < R

jryvnjqdx2andkunkLq(
)!0 and
p<q . We infer that Z

jrxv0jpdxlim inf
n!1Z

jrxvnjpdx= 0;
which, in view of inequality (2.5), implies v0= 0. But this is impossible, since kvnkLq(
)= 1, for each
n. Consequently, mshould be positive.
Claim 3. There exists u2N;u0a.e. in
, such that J(u) =m.
We begin by considering fungNa minimizing sequence, i.e. J(un)!masn!1 . By
Claim 2 , we have thatfungis a bounded sequence in W1;p;q
0(
). Thus, there exists u2W1;p;q
0(
), such
that, after eventually extracting a subsequence, unconverges weakly to uinW1;p;q
0(
) and strongly in
Lq(
). Moreover, un(x)!u(x) for a.e.x2
asn!1 . Standard lower semicontinuity arguments
imply that
J(u)lim inf
n!1J(un) =m:

2. Eigenvalue problems 15
Sinceun2Nfor alln2N, we have
Z

jrxunjp+Z

jryunjq=Z

junjq;8n: (2.10)
Ifu0 on
, thenR

junjq!0 asn!1 , and by (2.10), we obtainR

jrxunjp+R

jryunjq!0
asn!1 . Combining this with the fact that unconverges weakly to uinW1;p;q
0(
), we infer that,
actually,unconverges strongly to 0 in W1;p;q
0(
). Thus, we deduce that
0<Z

junjqZ

jryunjq=Z

jrxunjp!0:
Next, we can apply similar arguments as the one used in the proof of Claims 1 and 2in order to arrive
at a contradiction. Consequently, u60.
Now, letting n!1 in (2.10), we deduce
Z

jrxujp+Z

jryujqZ

jujq:
If we have equality in the above relation, then u2Nand everything is clear. Assume the contrary,
that is Z

jrxujp+Z

jryujq<Z

jujq: (2.11)
Letting
t=0
BB@Z

jujqdxZ

jryujqdx
Z

jrxujpdx1
CCA1
pq
;
thentu2N. Note also that t2(0;1), sincep<q and by (2.11),
Z

jujqZ

jryujq
Z

jrxujp>1:
Hence
0<mJ(tu) =1
p1
qZ

jrx(tu)jp
=tp1
p1
qZ

jrxujp
=tpJ(u)
tplim inf
n!1J(un) =tpm<m;
which represents a contradiction. Thus, we must haveR

jrxujp+R

jryujq=R

jujqand the proof
ofClaim 3 is clear.

2. Eigenvalue problems 16
We are now ready to complete the proof of Proposition 2.2. Let u2Nbe such that J(u) =m,
given by Claim 3 . Sinceu2N, we have
Z

jrxujp+Z

jryujq=Z

jujq;
and the fact that u60 impliesR

jryujq< R

jujq. Letv2W1;p;q
0(
) be arbitrary, but xed,
and let >0 be suciently small, so that for each s2(;), the function u+svdoes not vanish
everywhere in
and
Z

ju+svjq>Z

jry(u+sv)jq:
For eachs2(;), lett(s)>0 be, given by
t(s) :=0
BB@Z

ju+svjqZ
!jry(u+sv)jq
Z

jrx(u+sv)jp1
CCA1
pq
and note that, t(s)(u+sv)2N. Note that, function t(s) is dierentiable as the composition of
some dierentiable functions and since u2N, we havet(0) = 1. Next, dene
: (;)!Rby

(s) :=J(t(s)(u+sv)). By construction, we have
2C1(;) and
(0)
(s), for alls2(;).
Thus, we deduce that
0 =
0(0) =hJ0
(t(0)u);t0(0)u+t(0)vi=t0(0)hJ0
(u);ui+hJ0
(u);vi=hJ0
(u);vi:
The proof of Proposition 2.2 is complete. 
2.2 Fractional eigenvalue problems
In recent years increasing attention has been paid to the study of dierential and partial dierential
equations involving nonlocal operators, especially fractional Laplacian-type operators. The interest in
studying such problems was stimulated by their applications in continuum mechanics, phase transition
phenomena, population dynamics, image processing and game theory, see [15, 24, 59, 74] and the
references therein.
In this section,
RN(N2) will be a bounded domain with Lipschitz boundary @
. For
eachp2(1;1) ands2(0;1), we dene the nonlocal nonlinear operator, called the fractional (s;p)-
Laplacian , that is
(
p)su(x) := 2 p:v:Z
RNju(x)u(y)jp2(u(x)u(y))
jxyjN+spdy; x2RN: (2.12)
Forp= 2, the above denition reduces to the linear fractional Laplacian, (
)s.
Following the lines from [23, p. 1814], we recall that the natural setting for studying equations in-
volving the above operator, (
p)s, is the fractional Sobolev space Ws;p
0(RN), dened as the completion

2. Eigenvalue problems 17
ofC1
0(RN) with respect to the Gagliardo seminorm, dened as
[u]Ws;p(RN):=Z
RNZ
RNju(x)u(y)jp
jxyjN+spdxdy1
p
:
Considering the extension of uby 0 in RNn
we consider the space
fWs;p
0(
) :=n
u:RN!R: [u]Ws;p(RN)<+1andu= 0 in RNn
o
;
endowed with [
]Ws;p(RN). This is a re
exive Banach space. Moreover, since
is a bounded domain
with Lipschitz boundary, it is well known (see, e.g. [23, Proposition B.1]) that fWs;p
0(
) coincides with
the completion of C1
0(
) with respect to the norm [
]Ws;p(RN). Furthermore, fWs;p
0(
) is compactly
embedded in Lq(
), for each real number q2[1;p] (see, e.g. [40, Theorem 7.1]). More details regarding
this fractional spaces can be found, for instance, in the book [63], or in the papers [23, 35, 36, 40].
The common eigenvalue problem associated to the fractional ( s;p)-Laplace operator is given by
8
><
>:(
p)su(x) =ju(x)jp2u(x); x2
u(x) = 0; forx2RNn
:(2.13)
Dierent aspects concerning problem (2.13), have been investigated over time by Franzina & Palatucci
[53], Lindgren & Lindqvist [78], Brasco, Parini & Squassina [23], Del Pezzo & Quass [36] and Del Pezzo,
Fernandez Bonder & Lopez Rios [35]. It is well known that the rst eigenvalue of (2.13), denoted by
1(s;p), can be characterized from a variational point of view by
1(s;p) := inf
u2fWs;p
0(
)nf0g[u]p
Ws;p(RN)Z
RNjujpdx: (2.14)
Also, we recall that 1(s;p) is positive, simple and isolated (see e.g. [36, Theorems 4.9 and 4.11] and
[78, Theorems 14 and 19]). Moreover, its associated eigenfunctions never change sign in
.
In the next two subsections, we investigate the existence of solutions for two eigenvalue problems
associated to the fractional ( s;p)-Laplacian, dened above.
2.2.1 On an eigenvalue problem involving the fractional (s; p)-Laplacian
In this section, we are concerned with the following eigenvalue problem
8
><
>:(
p)su(x) =f(x;u(x));forx2
u(x) = 0; forx2RNn
;(2.15)
wheref:
R!Ris given by
f(x;t) =8
><
>:h(x;t);ift0;
jtjp2t;ift<0:(2.16)
We assume that h:
[0;1)!Ris a Caratheodory function satisfying the following hypotheses

2. Eigenvalue problems 18
(H1) there exists a positive constant C2(0;1) such thatjh(x;t)jCtp1, for anyt0 and a.e.
x2
;
(H2) there exists t0>0, such that H(x;t0) =Rt0
0h(x;s)ds> 0 for a.e.x2
;
(H3) lim
t!1h(x;t)
tp1= 0, uniformly in
.
Examples of functions h, which satisfy hypotheses (H1)-(H3), are given in [89], but we recall them here
for readers' convenience:
1.h(x;t) = sin((t=k)p1), for anyt0 and anyx2
, wherek>1 is a constant;
2.h(x;t) =klog(1 +tp1), for anyt0 and anyx2
, wherek2(0;1) is a constant;
3.h(x;t) =g(x)(tq(x)1tr(x)1), for anyt0 and anyx2
, whereq; r:
!(1;p) are continuous
functions satisfying max
r<min
qandg2L1(
) satises 0 <inf
gsup
g<1.
Note that, in the case when the fractional ( s;p)-Laplacian from the left-hand side of equation (2.15)
is replaced by a dierent type of fractional operators, equations of type (2.15) were studied in [7] and
[107]. Also, our study complements the results obtained by Mih ailescu & R adulescu [89] and Pucci &
R adulescu [96], in the case when in the left side of equation (2.15) dierential operators are considered,
namely the Laplace operator and the polyharmonic operator, respectively.
We dene an eigenvalue of (2.15) as being a real number , for which there exists a function
u2fWs;p
0(
)nf0g, such that
Es;p(u;v) =Z

f(x;u)v(x)dx;8v2fWs;p
0(
); (2.17)
where
Es;p(u;v) :=Z
RNZ
RNju(x)u(y)jp2(u(x)u(y))(v(x)v(y))
jxyjN+spdxdy;8u;v2fWs;p
0(
):
The main result of this section is given by the following theorem.
Theorem 2.2. (see [44, Theorem 2.1]) Assume that fis given by relation (2.16) and conditions (H1),
(H2) and (H3) are fullled. Then, 1(s;p)dened in (2.14), is an isolated eigenvalue of problem (2.15).
Moreover, any 2(0;1(s;p))is not an eigenvalue of problem (2.15), but there exists 1>  1(s;p),
such that any 2(1;1)is an eigenvalue of problem (2.15).
Proof of Theorem 2.2. For eachu2fWs;p
0(
) we set
u(x) = maxfu(x);0g;8×2
:
By [37, Lemma 2.2], we have that u+;u2fWs;p
0(
), while by [36, (4.29)] the following estimates hold
true
ju(x)u(y)jp2(u(x)u(y))(u+(x)u+(y))ju+(x)u+(y)jp;8x; y2
; (2.18)

2. Eigenvalue problems 19
and
ju(x)u(y)jp2(u(x)u(y))(u(x)u(y))ju(x)u(y)jp;8x; y2
: (2.19)
Thus, problem (2.15) with fgiven by relation (2.16), becomes
8
><
>:(
p)su(x) =(h(x;u+)up1
); x2
u(x) = 0; forx2RNn
:(2.20)
We dene an eigenvalue of (2.20) , a real number , such that there exists u2fWs;p
0(
)nf0ga weak
solution of problem (2.15), i.e.
Es;p(u;v) =Z


h(x;u+)up1

vdx;8v2fWs;p
0(
): (2.21)
Lemma 2.1. Any2(0;1(s;p))is not an eigenvalue of problem (2.20).
Proof. Assume that >0 is an eigenvalue of problem (2.20) with the corresponding eigenfunction
u. Takingv=u+andv=uin (2.21), we obtain
Es;p(u;u+) =Z

h(x;u+)u+dx (2.22)
and
Es;p(u;u) =Z

up
dx: (2.23)
By the denition of 1(s;p), we have
1(s;p)Z

jvjpdx[v]p
Ws;p(RN)=Es;p(v;v);8v2fWs;p
0(
): (2.24)
Taking into account relations (2.22), (2.24), (2.18), and hypothesis (H1), we have
1(s;p)Z

up
+dx[u+]p
Ws;p(RN)Es;p(u;u+) =Z

h(x;u+)u+dxZ

up
+dx;
while by relations (2.23), (2.24) and (2.19), we deduce
1(s;p)Z

up
dx[u]p
Ws;p(RN)Es;p(u;u) =Z

up
dx:
Sinceis an eigenvalue of problem (2.20), then u60 and by the above pieces of information, we
deduce that u+6= 0 oru6= 0. Hence, the last two inequalities show that is an eigenvalue of problem
(2.20), only if 1(s;p).
Lemma 2.2. The rst eigenvalue of (2.13), 1(s;p), is also an eigenvalue of problem (2.20).

2. Eigenvalue problems 20
Proof. As we have already pointed out, 1(s;p) is the lowest eigenvalue of problem (2.13), it
is simple and the corresponding eigenfunctions do not change sign in domain
. Then, there exists
u12fWs;p
0(
)nf0gwithu1(x)0, for each x2
, an eigenfunction corresponding to 1(s;p), i.e.
Es;p(u1;v) =1(s;p)Z

ju1jp2u1vdx =1(s;p)Z

(u1)p1vdx;
for anyv2fWs;p
0(
). Hence, we have ( u1)+= 0 and (u1)=u1and we deduce that relation (2.21)
holds true with u=u1and=1(s;p). Consequently, 1(s;p) is an eigenvalue of problem (2.20). The
proof of Lemma 2.2 is complete. 
Lemma 2.3. 1(s;p)is an isolated eigenvalue of problem (2.20).
Proof. By Lemma 2.1, we have that 1(s;p) is isolated in a neighborhood to the left. We will prove
that it is also isolated in a neighborhood to the right. To this aim, let 1(s;p) be an eigenvalue
of problem (2.20) with a corresponding eigenfunction u2fWs;p
0(
). If its corresponding positive part,
that isu+, is not identically zero, then by (2.24), (2.18) and hypothesis (H1), we deduce
1(s;p)Z

up
+dx[u+]p
Ws;p(RN)Es;p(u;u+) =Z

h(x;u+)u+dx
CZ

up
+dx:
SinceC2(0;1), then1(s;p)< 1(s;p)=C. It follows that, if 2(0;1(s;p)=C) is an eigenvalue
of problem (2.20), then it has a corresponding eigenfunction u2fWs;p
0(
) withu0 in
, or
Es;p(u;v) =Z

(u)p1vdx =Z

jujp2uvdx;8v2fWs;p
0(
):
It means that is an eigenvalue of problem (2.13), too. But we have already noted that 1(s;p) is an
isolated eigenvalue of problem (2.13), i.e. there exists >0, such that in the interval ( 1(s;p);1(s;p)+
) there is no eigenvalue of problem (2.13). Thus, taking := minf1(s;p)=C; 1(s;p) +g, we observe
that> 1(s;p) and any2(1(s;p);) cannot be an eigenvalue of problem (2.13), and consequently
any2(1(s;p);) is not an eigenvalue of problem (2.20). We conclude that 1(s;p) is an isolated
eigenvalue in the set of eigenvalues of problem (2.20). The proof of Lemma 2.3 is complete. 
In the following, we will show that there exists 1>0, such that any 2(1;1) is an eigenvalue
of problem (2.20). In order to do this, we consider the eigenvalue problem
8
><
>:(
p)su(x) =h(x;u+); x2
u(x) = 0; forx2RNn
:(2.25)
We say that a real number is an eigenvalue of (2.25) , if there exists u2fWs;p
0(
)nf0g, such that
Es;p(u;v) =Z

h(x;u+)v(x)dx;8v2fWs;p
0(
): (2.26)

2. Eigenvalue problems 21
We note that, if is an eigenvalue for (2.25) with the corresponding eigenfunction u, then testing with
v=uin the above relation, we deduce
Es;p(u;u) = 0;
or, by relation (2.19), we have that
[u]p
Ws;p(RN)Es;p(u;u) = 0;
which implies u= 0. Thus, we nd u0. In other words, the eigenvalues of problem (2.25)
possess only nonnegative corresponding eigenfunctions. Moreover, by the above facts, we deduce that
an eigenvalue of problem (2.25) is an eigenvalue of problem (2.20).
For each>0 we dene the energy functional associated to problem (2.25) by J:fWs;p
0(
)!R,
J(u) :=1
p[u]p
Ws;p(RN)Z

H(x;u+)dx;
whereH(x;t) =Rt
0h(x;s)ds. Standard arguments show that J2C1(fWs;p
0(
);R) with the derivative
given by
hJ0
(u);vi=Es;p(u;v)Z

h(x;u+)vdx;
for anyu;v2fWs;p
0(
). Observe that, in this context, >0 is an eigenvalue of problem (2.25), if and
only if there exists a nontrivial critical point of functional J.
Lemma 2.4. The functional Jis bounded from below and coercive.
Proof. By hypothesis (H3), we deduce that
lim
t!1H(x;t)
tp= 0;uniformly in
:
Then, for a given >0 there exists a positive constant C>0, such that
H(x;t)1(s;p)
2ptp+C;8t0;a.e.x2
:
Then, for each u2fWs;p
0(
), we have
J(u)1
p[u]p
Ws;p(RN)1(s;p)
2pZ

updxCj
j
1
2p[u]p
Ws;p(RN)Cj
j:
The last inequality shows that Jis bounded from below and coercive. The proof of Lemma 2.4 is
complete. 
Lemma 2.5. There exists 1>0, such that, assuming > 1, we have inffWs;p
0(
)J<0.

2. Eigenvalue problems 22
Proof. By hypothesis (H2), we deduce that there exists t0>0, such that H(x;t0)>0 for a.e.
x2
. Let
1
be a compact subset, suciently large, and u02C1
0(
)fWs;p
0(
), such that
u0(x) =t0, for anyx2
1and 0u0(x)t0, for anyx2
n
1.
Hence, by hypothesis (H1), we deduce
Z

H(x;u0)dxZ

1H(x;t0)dxZ

n
1Cup
0dx
Z

1H(x;t0)dxCtp
0j
n
1j>0:
Then, we infer that
J(u0)1
p[u0]p
Ws;p(RN)Z

1H(x;t0)dxCtp
0j
n
1j
<0;
for each
>1
p[u0]p
Ws;p(RN)Z

1H(x;t0)dxCtp
0j
n
1j:
Hence, we conclude that there exists 1>0, such that for any > 1, we have inf fWs;p
0(
)J<0. The
proof of Lemma 2.5 is complete. 
Taking into account Lemmas 2.4 and 2.5 and the fact that Jis weakly lower semi-continuous, by
Theorem 1.1, we deduce that there exists a constant 1>0, such that Jpossesses a negative global
minimum, for each > 1. It means that such a is an eigenvalue of problem (2.25) and consequently
an eigenvalue of problem (2.20). Combining these pieces of information with the results of Lemmas
2.1, 2.2 and 2.3, we conclude that Theorem 2.2 holds true. 
2.2.2 Perturbed fractional eigenvalue problems
The purpose of this section is to analyze the equation
8
><
>:(
p)su(x) + (
q)tu(x) =ju(x)jr2u(x);forx2
u(x) = 0; forx2RNn
;(2.27)
wheres,t,pandqare real numbers satisfying the assumption
0<t<s< 1;1<p<q<1; sN
p=tN
q; (2.28)
r2fp;qgand2Ris a parameter. Our goal is to determine all the parameters for which problem
(2.27) possesses nontrivial weak solutions. Note that, for p=qands=tproblem (2.27) reduces to
the eigenvalue problem for the fractional ( s;p)-Laplacian, that is equation (2.13). The problem that we
propose to investigate here represents a perturbation of problem (2.13) with a fractional ( t;q)-Laplacian

2. Eigenvalue problems 23
in the left-hand side. We point out that in the case when the nonlocal operators from equation (2.27)
are replaced by the corresponding dierential operators ( p-Laplacian and q-Laplacian) the resulting
problem was analyzed in [84, 48, 18, 85] under dierent boundary conditions.
Further, note that in problem (2.27) two nonlocal operators are involved, (
p)sand (
q)t,
respectively. The function space where we analyze problems involving (
p)swith homogeneous
Dirichlet boundary condition is the fractional Sobolev space fWs;p
0(
), while the function space where
we analyze problems involving (
q)twith homogeneous Dirichlet boundary condition is the fractional
Sobolev space fWt;q
0(
). Thus, in the situation which occurs from problem (2.27), we should decide which
of the spaces fWs;p
0(
) andfWt;q
0(
) is the natural function space where we can seek solutions for the
problem. A key condition in this case is assumption (2.28), which in view of Theorem 1.4.4.1 from [63],
assures that
fWs;p
0(
)fWt;q
0(
):
Moreover, this inclusion and [40, Theorem 7.1] assure that under assumption (2.28) the fractional
Sobolev space fWs;p
0(
) is compactly embedded in Lr(
) forr2fp;qg. It follows that the natural
function space where we should study equation (2.27) is fWs;p
0(
).
For simplicity, throughout this section, we will consider the notations
Es;p(u;v) :=Z
RNZ
RNju(x)u(y)jp2(u(x)u(y))(v(x)v(y))
jxyjN+spdxdy;
for allu; v2fWs;p
0(
) and
Et;q(u;v) :=Z
RNZ
RNju(x)u(y)jq2(u(x)u(y))(v(x)v(y))
jxyjN+tqdxdy;
for allu; v2fWs;p
0(
).
We dene a weak solution of (2.27 ) as being a function u2fWs;p
0(
), such that
Es;p(u;v) +Et;q(u;v) =Z

ju(x)jr2u(x)v(x)dx;8v2fWs;p
0(
): (2.29)
For eachs; t2(0;1) andp; q2(1;1), we consider
1(s;p) := inf
u2C1
0(
)nf0g[u]p
Ws;p(RN)Z
RNjujpdx;
and
1(t;q) := inf
u2C1
0(
)nf0g[u]q
Wt;q(RN)Z
RNjujqdx:
Thus, under assumption (2.28), we have
[u]p
Ws;p(RN)1(s;p)R

jujpdx;8u2fWs;p
0(
);
[u]q
Wt;q(RN)1(t;q)R

jujqdx;8u2fWs;p
0(
):(2.30)

2. Eigenvalue problems 24
Remark. It is known that 1(s;p)is attained at some u2fWs;p
0(
)nf0g(see [78, Theorem 5] or [53]
for a similar problem), with kukLp(RN)=kukLp(
)= 1and
[u]p
Ws;p(RN)Z
RNjujpdx=1(s;p):
Moreover, it holds true that
Es;p(u;') =1(s;p)Z
RNju(x)jp2u(x)'(x)dx;8'2fWs;p
0(
);
which means that 1(s;p)is an eigenvalue of problem (2.13). As it was pointed out in [78, p. 800],
Z
RNZ
RNju(x)u(y)jp
jxyjN+spdxdy =Z

Z

ju(x)u(y)jp
jxyjN+spdxdy
+ 2Z
RNn
dyZ

ju(x)jp
jxyjN+spdx;(2.31)
althoughu= 0inRNn
, and so
1(s;p)> inf
'2C1
0(
)nf0gZ

Z

j'(x)'(y)jp
jxyjN+spdxdy
Z

j'(x)jpdx:
Further, dene
1:=8
><
>:1(s;p);ifr=p
1(t;q);ifr=q:(2.32)
The main result of this section is given by the following theorem.
Theorem 2.3. (see [49, Theorem 1.1]) Assume condition (2.28) is fullled. Then the set of all real
parameters for which problem (2.27) has at least a nontrivial weak solution is the interval (1;1),
with1dened by relation (2.32). Moreover, the weak solution could be chosen to be non-negative.
Proof of Theorem 2.3. In the following, we will assume that condition (2.28) is fullled.
For eachs; t2(0;1),p; q2(1;1) andr2fp;qg, we dene
1:= inf
u2C1
0(
)nf0g8
>>>>>>>>><
>>>>>>>>>:1
p[u]p
Ws;p(RN)+1
q[u]q
Wt;q(RN)
1
pZ

jujpdx;ifr=p
1
p[u]p
Ws;p(RN)+1
q[u]q
Wt;q(RN)
1
qZ

jujqdx;ifr=q:
The following result plays an important role in our analysis.

2. Eigenvalue problems 25
Proposition 2.3. 1=1.
Proof. First, note that it is obvious that 11. Next, simple computations show that for each
u2C1
0(
)nf0gand each>0, we have
18
>>>>>>>>>><
>>>>>>>>>>:1
p[u]p
Ws;p(RN)+qp
q[u]q
Wt;q(RN)
1
pZ

jujpdx;ifr=p
pq
p[u]p
Ws;p(RN)+1
q[u]q
Wt;q(RN)
1
qZ

jujqdx;ifr=q:
Letting!0 ifr=por!1 ifr=qand passing to the inmum over u2C1
0(
)nf0gin the
right hand-side of the above relation we deduce that 11. The conclusion of this proposition is now
complete. 
Proposition 2.4. For each2(1;1], problem (2.27) has no nontrivial solution.
Proof. First, note that assuming that for some 0 problem (2.27) has a nontrivial weak solution,
sayu, then testing in relation (2.29) with v=u, we arrive at a contradiction. Thus, for any parameter
2(1;0], problem (2.27) does not have nontrivial weak solutions.
Next, let2(0;1). Assume by contradiction that there exists u2fWs;p
0(
)nf0g, a weak solution
of problem (2.27). By the denitions of 1,1(s;p) and1(t;q) and taking v=uin (2.29), we get
0<8
>><
>>:1(s;p)
pZ

jujpdx1
p[u]p
Ws;p(RN)
pZ

jujpdx0;ifr=p
1(t;q)
qZ

jujqdx1
q[u]q
Wt;q(RN)
qZ

jujqdx0;ifr=q;
which represents a contradiction. It follows that for any parameter 2(0;1), problem (2.27) does not
possess nontrivial weak solutions.
In order to complete the proof of the proposition, we will show that 1cannot be an eigenvalue of
problem (2.27). Indeed, if we assume again by contradiction that there exists u2fWs;p
0(
)nf0gsuch
that (2.27) holds with =1, then letting v=uin (2.29) and using the estimates (2.30) we deduce
[u]p
Ws;p(RN)+ [u]q
Wt;q(RN)=8
>><
>>:1(s;p)Z

jujpdx[u]p
Ws;p(RN);ifr=p
1(t;q)Z

jujqdx[u]q
Wt;q(RN);ifr=q:
The above estimates imply either [ u]Wt;q(RN)= 0, ifr=p, or [u]Ws;p(RN)= 0, ifr=q, which combined
with relations (2.30) yield that either kukLq(
)= 0, ifr=p, orkukLp(
)= 0, ifr=q. It follows that
u= 0, which represents a contradiction. Thus, for =1, problem (2.27) does not possess nontrivial
solutions and the proof of this proposition is complete. 

2. Eigenvalue problems 26
Proposition 2.5. For each2(1;1), problem (2.27) has a nontrivial solution.
In order to prove Proposition 2.5, we start by dening for each > 1the so-called energy functional
associated to problem (2.27) as I:fWs;p
0(
)!R, given by
I(u) =1
p[u]p
Ws;p(RN)+1
q[u]q
Wt;q(RN)
rZ

jujrdx:
Standard arguments can be used in order to deduce that I2C1(fWs;p
0(
);R) with the derivative given
by
hI0
(u);vi=Es;p(u;v) +Et;q(u;v)Z

jujr2uvdx;8u; v2fWs;p
0(
):
We note that problem (2.27) possesses a nontrivial weak solution for a certain , if and only if I
possesses a non-trivial critical point. We consider the Nehari manifold
N:=n
u2fWs;p
0(
)nf0g:hI0
(u);ui= 0o
=
u2fWs;p
0(
)nf0g: [u]p
Ws;p(RN)+ [u]q
Wt;q(RN)=Z

jujrdx
:
Note that for each u2N, the functional Ihas the following expression
I(u) =[u]p
Ws;p(RN)
p+[u]q
Wt;q(RN)
q
rZ

jujrdx
=8
>><
>>:1
q1
p
[u]q
Wt;q(RN)0;ifr=p
1
p1
q
[u]p
Ws;p(RN)0;ifr=q;(2.33)
and the following inequalities hold true
8
>><
>>:Z

jujpdx> [u]p
Ws;p(RN);ifr=p
Z

jujqdx> [u]q
Wt;q(RN);ifr=q:(2.34)
Lemma 2.6.N6=;.
Proof. Since> 1, we deduce that there exists w2fWs;p
0(
)nf0gfor which either
Z

jwjpdx> [w]p
Ws;p(RN);ifr=p;
or
Z

jwjqdx> [w]q
Wt;q(RN);ifr=q:
Then, there exists >0 such that w2N, i.e.
p[w]p
Ws;p(RN)+q[w]q
Wt;q(RN)=8
>><
>>:pZ

jwjpdx; ifr=p
qZ

jwjqdx; ifr=q;

2. Eigenvalue problems 27
which is obvious with
=8
>>>>>>>>>><
>>>>>>>>>>:0
B@Z

jwjpdx[w]p
Ws;p(RN)
[w]q
Wt;q(RN)1
CA1
qp
;ifr=p
0
B@Z

jwjqdx[w]q
Wt;q(RN)
[w]p
Ws;p(RN)1
CA1
pq
;ifr=q:

Set
m:= inf
v2NI(v):
By (2.33,) we deduce that I(v)0, for allv2N, ifr=qandI(v)<0, for allv2N, ifr=p.
Thus,m0, ifr=qandm<0, ifr=p. We will show that mcan be achieved on N. The cases
r=qandr=pwill be analyzed separately.
Lemma 2.7. Ifr=q, then every minimizing sequence of Iis bounded in fWt;q
0(
)andfWs;p
0(
).
Proof. LetfungnNbe a minimizing sequence for I, i.e.
[un]p
Ws;p(RN)!1
p1
q1
m;asn!1:
First, we will prove thatn
[un]Wt;q(RN)o
nis a bounded sequence. Assume the contrary, that is
[un]Wt;q(RN)!1;asn!1:
Since for each nwe haveun2N, we deduce thatR

junjqdx!1 , asn!1 .
Setwn:=un=kunkLq(
). Since inequality (2.34) holds true for any n, we deduce that [ wn]q
Wt;q(RN)<
for anyn. Thus, the sequence fwngnis bounded in fWt;q
0(
).
Next, since the sequence f[un]p
Ws;p(RN)gnis bounded and the sequence fR

junjqdxgnis unbounded,
we get
[wn]p
Ws;p(RN)=[un]p
Ws;p(RN)
kunkp
Lq(
)!0;asn!1:
By the above relation, we deduce that fwngnis bounded in fWs;p
0(
). Consequently, there exists
w2fWs;p
0(
) such that wnconverges weakly to winfWs;p
0(
) andwnconverges strongly to winLq(
).
Thus,
[w]p
Ws;p(RN)lim inf
n!1[wn]p
Ws;p(RN)= 0;
which implies that w= 0. On the other hand, since kwnkLq(
)= 1 for each n, we get thatkwkLq(
)= 1,
which is a contradiction with w= 0. Hence, the sequence f[un]Wt;q(RN)gnis bounded, or unis bounded

2. Eigenvalue problems 28
infWt;q
0(
). By (2.30), we deduce that unis bounded in Lq(
), too. Finally, taking into account the
fact thatfungnN, it follows thatf[un]Ws;p(RN)gnis bounded, or unis bounded in fWs;p
0(
). The
proof of Lemma 2.7 is now complete. 
Lemma 2.8. Ifr=q, thenm>0.
Proof. We already observed that m0. Suppose by contradiction that m= 0. LetfungnN
be a minimizing sequence for m= 0, that is
[un]p
Ws;p(RN)!0;asn!1:
Then, by Lemma 2.7 we deduce that the sequence fungnis bounded in fWs;p
0(
) andfWt;q
0(
). Thus,
there exists u2fWs;p
0(
), such that unconverges weakly to uinfWs;p
0(
) andfWt;q
0(
) andunconverges
strongly to uinLq(
). Standard lower semicontinuity arguments, imply that
[u]p
Ws;p(RN)lim inf
n!1[un]p
Ws;p(RN)= 0:
Thusu= 0. Consequently, unconverges weakly to 0 in fWs;p
0(
) andfWt;q
0(
) andR

junjqdx!0, as
n!1 .
Next, letwn=un=kunkLq(
). Taking into account that inequality (2.34) holds true for each n, we
obtain that [ wn]q
Wt;q(RN)<, for anyn. Consequently, wnis bounded in fWt;q
0(
). On the other hand,
since for each n, we haveun2Nandp<q , we obtain that
[wn]p
Ws;p(RN)=kunkqp
Lq(
)
[wn]q
Wt;q(RN)
!0;asn!1;
The above pieces of information yield that wnis bounded in fWs;p
0(
) andfWt;q
0(
). It follows that there
existsw2fWs;p
0(
) such that wnconverges weakly to winfWs;p
0(
) andfWt;q
0(
) andwnconverges
strongly to winLq(
). Thus,R

jwjqdx= 1 and
[w]p
Ws;p(RN)lim inf
n!1[wn]p
Ws;p(RN)= 0:
The last relation, in view of relation (2.30), implies that w= 0. But this is a contradiction with the
fact thatkwkLq(
)= 1. Consequently, mshould be positive. The proof of Lemma 2.8 is complete. 
Lemma 2.9. Ifr=q, then there exists u2N, such that I(u) =m.
Proof. LetfungnNbe a minimizing sequence for m, that isI(un)!masn!1 . By
Lemma 2.7, we have that unis bounded in fWs;p
0(
) andfWt;q
0(
). Thus, there exists u2fWs;p
0(
), such
thatunconverges weakly to uinfWs;p
0(
) andfWt;q
0(
) andunconverges strongly to uinLq(
).
By the above pieces of information, we deduce that
I(u)lim inf
n!1I(un) =m:
Sinceun2Nfor eachn, we have
[un]p
Ws;p(RN)+ [un]q
Wt;q(RN)=Z

junjqdx; for anyn: (2.35)

2. Eigenvalue problems 29
Ifu= 0 in
, then R

junjqdx!0 asn!1 and by the above relation, we deduce that [ un]p
Ws;p(RN)+
[un]q
Wt;q(RN)!0 asn!1 . It follows that I(un)!0, asn!1 , which represents a contradiction
with the fact that I(un)!m>0 asn!1 . Consequently, u6= 0.
Next, letting n!1 in relation (2.35), we get
[u]p
Ws;p(RN)+ [u]q
Wt;q(RN)Z

jujqdx: (2.36)
Assume by contradiction that in (2.36) the strict inequality holds, i.e.
[u]p
Ws;p(RN)+ [u]q
Wt;q(RN)<Z

jujqdx: (2.37)
Let>0 be, such that u2N, that is
=0
BB@Z

jujqdx[u]q
Wt;q(RN)
[u]p
Ws;p(RN)1
CCA1
pq
:
Note that, since p<q and (2.37) holds true, we get 2(0;1).
Finally, since u2Nand2(0;1), we have
0<mI(u) =1
p1
q
[u]p
Ws;p(RN)
=p1
p1
q
[u]p
Ws;p(RN)
=pI(u)
plim inf
n!1I(un) =pm<m;
which represents a contradiction. Thus, inequality (2.37) cannot hold true. Therefore, u2Nand by
I(u)m, it follows that I(u) =m. The proof of the lemma is now complete. 
Lemma 2.10. Ifr=p, thenNis bounded in fWs;p
0(
)andfWt;q
0(
).
Proof. First, we show that if fungnNthenf[un]p
Ws;p(RN)gnis a bounded sequence. Assume by
contradiction that [ un]Ws;p(RN)!1 , asn!1 . Next, letwn:=un=[un]Ws;p(RN). Then [wn]Ws;p(RN)=
1 for eachn, which means that fwngnis bounded in fWs;p
0(
). Thus, there exists w2fWs;p
0(
) such
that
wn* w infWs;p
0(
);
wn!winLp(
);
wn(x)!w(x) a.e. in
:
Sinceun2N, for eachn, it follows that (2.34) holds true with r=pfor eachun, and we deduce that
R

jwnjpdx> 1, for each n. Passing to the limit as n!1 , we obtain that
Z

jwjpdx1: (2.38)

2. Eigenvalue problems 30
On the other hand, since un2Nandp<q , we have
[wn]q
Wt;q(RN)= [un]pq
Ws;p(RN)
Z

jwnjpdx1
!0;asn!1:
Thus,wnconverges strongly to 0 in Lq(
). In particular, this means that wn(x)!0 for a.e. in
,
and consequently w= 0, which contradicts (2.38). It follows thatn
[un]Ws;p(RN)o
nis bounded provided
thatfungnN. By relation (2.30), we deduce that fR

junjpdxgnis a bounded sequence, too. Since
un2Nfor eachn, we deduce that the sequencen
[un]Wt;q(RN)o
nis bounded, and thus, the proof of
Lemma 2.10 is complete. 
Lemma 2.11. Ifr=p, thenm2(1;0).
Proof. We already observed that m<0. We will show that m6=1. By Lemma 2.10 there
exists a positive constant Msuch that [u]p
Ws;p(RN)Mand [u]q
Wt;q(RN)M, for eachu2N. Then,
sincep<q , we have
I(u) =1
q1
p
[u]q
Wt;q(RN)1
q1
p
M >1:
Thus,Iis bounded from below on N, which implies that m6=1. The proof of the lemma is
complete. 
Lemma 2.12. Ifr=p, then there exists u2N, such that I(u) =m.
Proof. LetfungnNbe a minimizing sequence for IonN, i.e.
I(un) =1
q1
p
[un]q
Wt;q(RN)!masn!1:
By Lemma 2.10, we have that Nis bounded in fWs;p
0(
) andfWt;q
0(
), and we deduce that there exists
u2fWs;p
0(
) such that unconverges weakly to uinfWs;p
0(
) (andfWt;q
0(
)) andunconverges strongly
touinLp(
). Then
[u]p
Ws;p(RN)lim inf
n!1[un]p
Ws;p(RN);
and
Z

junjpdx!Z

jujpdxasn!1:
Thus,1
p1
q
[u]p
Ws;p(RN)Z

jujpdx
lim inf
n!11
p1
q

[un]p
Ws;p(RN)Z

junjpdx
= lim inf
n!1I(un) =m<0:(2.39)
Hence
[u]p
Ws;p(RN)<Z

jujpdx;

2. Eigenvalue problems 31
and thenu6= 0.
Using the fact that un2Nfor everyn, we have
[un]p
Ws;p(RN)+ [un]q
Wt;q(RN)=Z

junjpdx;8n:
Passing to the limit as n!1 in the above relation and taking into consideration that unconverges
weakly touinfWs;p
0(
) andfWt;q
0(
) and strongly in Lp(
), we get
[u]p
Ws;p(RN)+ [u]q
Wt;q(RN)Z

jujpdx: (2.40)
If we have equality in the above relation then u2Nand everything is clear. Assume by contradiction
that in (2.40) the strict inequality holds, i.e.
[u]p
Ws;p(RN)+ [u]q
Wt;q(RN)<Z

jujpdx: (2.41)
Taking
:=0
BB@Z

jujpdx[u]p
Ws;p(RN)
[u]q
Wt;q(RN)1
CCA1
qp
;
we haveu2N, and it follows by (2.41) that >1. Then
I(u) =1
p1
q
p
[u]p
Ws;p(RN)Z

jujpdx
<1
p1
q
[u]p
Ws;p(RN)Z

jujpdx
=I(u)
lim inf
n!1I(un) =m;
which represents a contradiction. Thus inequality (2.41) cannot hold true. Therefore, in (2.40) only
the equality holds true, which means that u2N. By (2.39), it follows that I(u)m, and thus
I(u) =m. The proof of the lemma is now complete. 
We are now ready to complete the proof of Proposition 2.5. Let u2Nbe such that I(u) =m,
(whereuis given by Lemma 2.9, if r=qand by Lemma 2.12, if r=p). Sinceu2N, we have
[u]p
Ws;p(RN)+ [u]q
Wt;q(RN)=Z

jujrdx;
and the fact that u60 implies [u]p
Ws;p(RN)< R

jujp, ifr=pand [u]q
Wt;q(RN)< R

jujq, if
r=q. Letv2fWs;p
0(
) be arbitrary but xed, and let  >0 be suciently small, so that for each
2(;) the function u+vdoes not vanish everywhere in
and either
Z

ju+vjpdx> [u+v]p
Ws;p(RN);ifr=p;

2. Eigenvalue problems 32
or
Z

ju+vjqdx> [u+v]q
Wt;q(RN);ifr=q:
For each2(;), let()>0 be given either by
() :=0
BB@Z

ju+vjpdx[u+v]p
Ws;p(RN)
[u+v]q
Wt;q(RN)1
CCA1
qp
;ifr=p;
or by
() :=0
BB@Z

ju+vjqdx[u+v]q
Wt;q(RN)
[u+v]p
Ws;p(RN)1
CCA1
pq
;ifr=q:
Note that in both situations we have ()(u+v)2Nand function () is dierentiable as the
composition of some dierentiable functions. On the other hand, since u2Nwe have(0) = 1.
Next, dene
: (;)!Rby
() :=I(()(u+v)). By construction, we have
2C1(;)
and
(0)
(), for all2(;). Thus, we deduce that
0 =
0(0) =hI0
((0)u);0(0)u+(0)vi
=0(0)hI0
(u);ui+hI0
(u);vi=hI0
(u);vi:
Sincev2fWs;p
0(
) was arbitrary, we deduce that the last relation holds true for each v2fWs;p
0(
)
and, thus,uis a nontrivial critical point of Iand consequently a nontrivial weak solution of equation
(2.27). The proof of Proposition 2.5 is complete. 
Proposition 2.6. Ifu2Nis the minimizer of IoverN, i.e.I(u) =m, given by Lemmas 2.9
(in the case r=q) or 2.12 (in the case r=p), thenjujis a minimizer of IoverN, too.
Proof. It is well-known that for every a,b2Rthe following relations hold true
jabjjjajjbjj;
and
jabj>jjajjbjj;if ab< 0:
Using these two inequalities, we deduce that for each '2fWs;p
0(
) and each x; y2RNwe have
j'(y)'(x)jjj'(y)jj'(x)jj;
and
j'(y)'(x)j>jj'(y)jj'(x)jj;if '(y)'(x)<0:
It follows that
[j'j]p
Ws;p(RN)[']p
Ws;p(RN);and [j'j]q
Wt;q(RN)[']q
Wt;q(RN);8'2fWs;p
0(
):

2. Eigenvalue problems 33
Thus, we obtain
I(juj)I(u): (2.42)
Further, we split the discussion in two cases.
Assumer=q. Ifjuj2Neverything is clear. Let us assume by contradiction that juj62N. Then,
the above pieces of information assure that
[juj]p
Ws;p(RN)+ [juj]q
Wt;q(RN)<Z

jujqdx:
Let>0 be, such that juj2N, that is
=0
BB@Z

jujqdx[juj]q
Wt;q(RN)
[juj]p
Ws;p(RN)1
CCA1
pq
:
Sincep<q , we get2(0;1). On the other hand, the following relations hold
m=I(u)I(juj) =p1
p1
q
[u]p
Ws;p(RN)=pI(u)<I(u) =m;
which represents a contradiction. Thus, we must have juj2N, which forcesjujto be a minimizer of
IoverN.
Assumer=p. Ifjuj2N, then using the fact that q>p , we have
I(juj) =1
q1
p
[juj]q
Wt;q(RN)1
q1
p
[u]q
Wt;q(RN)=I(u) =m:
This estimate and relation (2.42) yield I(juj) =I(u) =mand everything is clear. Let us assume by
contradiction that juj62N. Then the above pieces of information assure that
[juj]p
Ws;p(RN)+ [juj]q
Wt;q(RN)<Z

jujpdx:
Let>0, be such that juj2N, that is
=0
BB@Z

jujpdx[juj]p
Ws;p(RN)
[juj]q
Wt;q(RN)1
CCA1
qp
:
Sincep<q , we get2(1;1). On the other hand, the following relations hold
mI(juj) =1
q1
p
q[juj]q
Wt;q(RN)
=1
p1
q
p
[juj]p
Ws;p(RN)Z

jujpdx
<1
p1
q
[juj]p
Ws;p(RN)Z

jujpdx
1
p1
q
[u]p
Ws;p(RN)Z

jujpdx
=I(u) =m;

2. Eigenvalue problems 34
which represents a contradiction. Thus, we must have, again, juj2N. It follows thatjujis a minimizer
ofIoverN.
Finally, by Propositions 2.4, 2.5 and 2.6 we obtain the conclusion of Theorem 2.3. 
2.3 Continuity of the rst eigenvalue for a family of degenerate eigen-
value problems
In this section, we consider that 0 2
. For each real number 2[0;2), consider the eigenvalue problem
8
><
>:div(jxjru) =u; forx2
u= 0; forx2@
;(2.43)
and the Rayleigh quotient corresponding to this equation
Z

jxjjruj2dx
Z

u2dx: (2.44)
The inmum of the above quotient among all smooth functions with zero boundary values, i.e.
1() := inf
u2C1
0(
)nf0gZ

jxjjruj2dx
Z

u2dx;
is positive and gives the rst eigenvalue of problem (2.43) (see Caldiroli & Musina [26, Theorem 4.1]).
Thus, we can dene the function 1: [0;2)!(0;1). The goal of this section is to show that function
1is continuous.
In the case when = 0, the dierential operator from problem (2.43), reduces to the Laplace
operator. If >0, the potentialjxjfrom the divergence operator vanishes at the origin and leads to
an operator which is not strictly elliptic (in the sense pointed out by Gilbarg & Trudinger in [58, p. 31])
on domains containing the origin of the Euclidian space. Problems involving such kind of degenerate
operators, received a great deal of attention, since they can model with sucient accuracy several
physical phenomena related to equilibrium of anisotropic continuous media (see Dautray & Lions [32]).
On the other hand, note that when  > 0 the classical Sobolev space H1
0(
) used in the study of
variational problems involving the Laplace operators is no longer adequate to the new situation and
consequently, we have to introduce a new functional setting for degenerate problems as (2.43), which
is pointed out in [25, Section 1] and [26, Section 3]. To be more precise, we let D1
0(
;jxj) be the
completion of the C1
0(
) under the norm
kuk2
=Z

jxjjruj2dx:

2. Eigenvalue problems 35
Actually,D1
0(
;jxj) is a Hilbert space with respect to the scalar product
hu;vi:=Z

jxjrurvdx:
Note that, in the case when = 0, the spaceD1
0(
;jxj0) reduces to the classical Sobolev space H1
0(
).
By [26, Proposition 3.2 and Remark 3.3], we deduce that D1
0(
;jxj) is compactly embedded in L2(
),
if and only if 2[0;2). Note that for = 2 this embedding fails to be compact.
Turning back to problem (2.43), we say that 2Ris an eigenvalue of problem (2.43), if there exists
u2D1
0(
;jxj)nf0g, such that
Z

jxjrurvdx =Z

uvdx;8v2D1
0(
;jxj):
When2[0;2), we can apply to problem (2.43) the standard spectral theory for compact, self-adjoint
operators (see, for example Willem [106, Chapter 3.4]), in order to describe the spectrum of this problem
as being an increasing, unbounded sequence of positive eigenvalues. Moreover, by [26, Theorem 4.1],
we know that for each 2[0;2), we have 1()>0 and it is achieved in D1
0(
;jxj) by a non negative
and unique (up to a multiplicative constant) function u, which veries
Z

jxjrurvdx =1()Z

uvdx;8v2D1
0(
;jxj):
Consequently, 1() is an eigenvalue of problem (2.43), actually the rst eigenvalue of the problem.
The main result of this section is given by the following theorem.
Theorem 2.4. (see [45, Theorem 1]) The function 1: [0;2)!(0;1)is continuous.
Proof of Theorem 2.4. Let02[0;2) be a real number and let fng[0;2) be a sequence,
such thatn!0, asn!1 . Our goal is to show that 1(n)!1(0), asn!1 .
First, we show that we can reduce our analysis to the case when
is a subset of the unit ball
centered at the origin from RN, i.e.,
B1(0), whereB1(0) =fx2RN:jxj<1g.
To this aim, for each >0, dene the rescaled domain

:=1
=fx2RN:y=x2
g:
Since 02
, it is clear that 0 2
, for all>0. Next, for each u:
!R, denev:
!Rby
v(x) =u(x);8×2
:
Note thatu2D1
0(
;jxj) impliesv2D1
0(
;jxj). Actually, a simple change of variable shows that
Z

jxjjruj2dx=+N2Z

jxjjrvj2dx;

2. Eigenvalue problems 36
and Z

u2dx=NZ

v2dx:
Consequently, we haveZ

jxjjruj2dx
Z

u2dx=2Z

jxjjrvj2dx
Z

v2dx:
From the denition of 1(), it follows that
1(;
) =21(;
);8>0:
In the above relation, we denoted by 1(;
) and1(;
) the minimum of the Rayleigh quotient
given by (2.44) among all smooth functions with zero boundary values from
and
, respectively.
This relation shows that 1(n;
)!1(0;
), asn!1 , if and only if 1(n;
)!1(0;
), as
n!1 , for all >0. Thus, we can reduce the analysis of the continuity of function 1to the case
when
B1(0).
Note that, if
B1(0), then for each two real numbers 1and2, such that 012<2, we
have Z

jxj2jruj2dxZ

jxj1jruj2dx;8u2D1
0(
;jxj1): (2.45)
In particular, this relation shows that D1
0(
;jxj1)D1
0(
;jxj2), if 012<2.
Next, we establish, via De Giorgi's -convergence, the asymptotic behavior of the sequence of energy
functionals associated to the dierential operator involved in problems of type (2.43).
For each positive integer n, consider the family of functionals In:L2(
)![0;1], dened by
In(u) =8
><
>:Z

jxjnjruj2dx; ifu2D1
0(
;jxjn);
+1; otherwise:
Theorem 2.5. DeneI:L2(
)![0;1]by
I(u) =8
><
>:Z

jxj0jruj2dx; ifu2D1
0(
;jxj0);
+1; otherwise:
Then lim
n!1In=I.
Proof. Letu02L2(
) andfungL2(
) be such that un!u0inL2(
). If we have lim inf
n!1In(un) =
1the conclusion is obvious. Assume that fungD1
0(
;jxjn) is such that
lim inf
n!1In(un) = lim
n!1In(un) =:L<1:

2. Eigenvalue problems 37
Let>0 be arbitrary, but xed, such that 0+<2. Sincen!0, asn!1 , there exists N2N
such that
n< 0+<2;8nN:
Thus, for all nN, by relation (2.45), we have that unD1
0(
;jxjn)D1
0(
;jxj0+) and
Z

jxj0+jrunj2dxZ

jxjnjrunj2dx<1:
Hence,fungnis bounded inD1
0(
;jxj0+) and there exists u2D1
0(
;jxj0+) such that unconverges
weakly touinD1
0(
;jxj0+), for each>0 small enough. The compact embedding of D1
0(
;jxj0+)
intoL2(
) assures that un!uinL2(
). Since un!u0inL2(
) we deduce that u0=uand
consequently, u02D1
0(
;jxj0+), for each>0 small enough. It follows that
Z

jxj0+jru0j2dxlim inf
n!1Z

jxj0+jrunj2dx
lim inf
n!1Z

jxjnjrunj2dx<1;for all>0 small enough :(2.46)
Thus, sup>0R

jxj0+jru0j2dx<1and by Fatou's lemma, jxj0jru0j2= lim inf !0jxj0+jru0j22
L1(
) andZ

jxj0jru0j2dxlim inf
!0Z

jxj0+jru0j2dx:
Combining the above pieces of information with the fact that we assumed @
to be smooth enough, we
can deduce (with similar arguments as those used in [21, Chapter 9.4] in the case of classical Sobolev
spaces) that u02D1
0(
;jxj0).
The last inequality combined with relation (2.46), yields
I(u0)lim inf
n!1In(un):
It remains to prove the existence of a recovery sequence for the -limit. Let u02L2(
). Note, if
u0=2D1
0(
;jxj0), there is nothing to prove, since I(u0) =1in this case. Assume that u02D1
0(
;jxj0)
and considerfungn2C1
0(
) be such that un!u0asn!1 inD1
0(
;jxj0). By the above fact, we
deduce that
lim
n!1I(un) =I(u0): (2.47)
For eachm2Nxed, by Lebesgue's dominated convergence theorem, we have
lim
n!1Z

jxjnjrumjdx=Z

jxj0jrumjdx;
which is equivalent with
lim
n!1In(um) =I(um): (2.48)
Taking into account relations (2.47) and (2.48) and applying Proposition 1.1 for X=L2(
),Fn=In
andF=Iwe obtain
I(u0)lim sup
n!1In(un):
The proof of Theorem 2.5 is complete. 

2. Eigenvalue problems 38
Remark 2.1. Note that, in the case when @
is not smooth enough, we can expect to encounter a
phenomenon similar with that pointed out by Lindqvist in [80], when the continuity of the rst eigenvalue
of thep-Laplacian fails to be valid.
Finally, we use Theorem 2.5 to establish the continuity of function 1, with respect to on [0;2).
First, we prove that
1(0)lim sup
n!11(n): (2.49)
By [26, Theorem 4.1], we know that there exists u2D1
0(
;jxj0), such thatR

jxj0jruj2dx=1(0)
andR

u2dx= 1. Letfungn2C1
0(
) be such that un!uasn!1 inD1
0(
;jxj0), i.e.
lim
n!1Z

jxj0jrunj2dx=Z

jxj0jruj2dx:
As in the second part of the proof of Theorem 2.5, for each m2Nthe constant sequence fumgnsatises
lim
n!1Z

jxjnjrumj2dx=Z

jxj0jrumj2dx:
Taking into account the above two relations, we can apply Proposition 1.1 with X=L2(
),Fn(u) =R

jxjnjruj2dxandF(u) =R

jxj0jruj2dx. Thus,fungis a recovery sequence and satises
lim sup
n!1Z

jxjnjrunj2dxZ

jxj0jruj2dx=1(0): (2.50)
SinceD1
0(
;jxj0) is compactly embedded in L2(
), we deduce that unconverges strongly to uin
L2(
), that is
lim
n!1Z

u2
ndx=Z

u2dx= 1:
By the denition of 1(n), we have
Z

jxjnjrunj2dx1(n)Z

junj2dx;8n1: (2.51)
SinceR

u2dx= 1, we deduce that for all  > 0 suciently small, there exists N2Nsuch thatR

u2
ndx> 1, for allnN. Then,1(n)R

u2
ndx> 1(n)(1), for allnN. Lettingn!1
in (2.51) and taking into account relation (2.50), we get
1(0)(1) lim sup
n!11(n);for all>0 small enough :
Taking!0 in the above relation, we conclude that
1(0)lim sup
n!11(n):
Next, we show that
1(0)lim inf
n!11(n): (2.52)

2. Eigenvalue problems 39
To this aim, we consider fvngD1
0(
;jxjn), such thatR

jxjnjrvnj2dx=1(n) andR

v2
ndx= 1.
Sincen!0, for >0 xed, suciently small, there exists N2Nsuch thatn<  0+, for all
nN. Then, by relation (2.45), we have
Z

jxj0+jrvnj2dxZ

jxjnjrvnj2dx<1;8nN:
Takingn!1 in this relation and using the fact that relation (2.49) holds true, we get
lim sup
n!1Z

jxj0+jrvnj2dxlim inf
n!1Z

jxjnjrvnj2dx
= lim inf
n!11(n)
lim sup
n!11(n)1(0):
By the above estimates, we deduce that fvngis bounded inD1
0(
;jxj0+) and thus, there exists
v2D1
0(
;jxj0+), such that vnconverges weakly to vinD1
0(
;jxj0+). Furthermore, the compact
embedding ofD1
0(
;jxj0+) intoL2(
) assures that vn!vinL2(
). In particular, it follows that
kvkL2(
)= 1. On the other hand, by Theorem 2.5 we deduce that we can apply Proposition 1.2 with
X=L2(
),Fn(u) =R

jxjnjruj2dx,F(u) =R

jxj0jruj2dx,zn=vnandz=vin order to obtain
that
1(0) =Z

jxj0jrvj2dx= lim inf
n!1Z

jxjnjrvnj2dx= lim inf
n!11(n):
Relations (2.49) and (2.52) imply the conclusion of Theorem 2.4. 
2.4 A maximum principle for a class of rst order dierential oper-
ators
One of the basic tools involved in the study of PDE's is the maximum principle . The importance of the
maximum principles is mainly due to the fact that it enables us to obtain information about dierential
equations and inequalities without any explicit knowledge of the solutions themselves (according to Pucci
& Serrin [97, p. IX]). The most common statement of the maximum principle leads us to think back
to the classical maximum principle for elliptic operators, which reads as follows (see, e.g. Gilbarg &
Trudinger [58, Theorem 3.1], Pucci & Serrin [97, Theorem 2.1.1] or Protter & Weinberger [95, Chapter
2]):
Theorem 2.6. LetLbe a linear elliptic dierential operator in
. Suppose that
Lp0 (0);in
; (2.53)
withp2C2(
)\C(
). Then the maximum (minimum) of pin
is achieved on @
, that is
max
x2
p(x) = max
x2@
p(x) (min
x2
p(x) = min
x2@
p(x)):

2. Eigenvalue problems 40
Remark 1. a. As it is pointed out in [97] the hypothesis that pis twice dierentiable is essential for
Theorem 2.6 .
b. The prototype dierential operator for which Theorem 2.6 holds true is the Laplace operator , i.e.

p=NX
i=1@2p
@x2
i:
c. If we replace condition (2.53) by
Lp+c(x)p0 (0);in
; (2.54)
withc(x)0 in
, then the conclusion of Theorem 2.6 remains valid, if we assume that p0 in
(p0 in
).
A natural question can be considered in connection with the classical maximum principle for elliptic
operators :\Does there exist another class of operators (dierent from elliptic operators)
for which the conclusion of Theorem 2.6 holds true?" .
In the following, we will give a positive answer to the above question, but rst, we present an
eigenvalue problem for the p(
)-Laplace operator, which represents the starting point of our study.
AssumeN1 and letp:
!(1;1) be a continuous function satisfying inf x2
p(x)>1 and
supx2
p(x)<1. Dene the p(
)-Laplace operator by

p(
)u:= div(jrujp(
)2ru):
Consider the eigenvalue problem
8
><
>:
p(x)u=jujp(x)2u;forx2
u= 0; forx2@
:(2.55)
We say that 2Ris an eigenvalue of problem (2.55) if there exists a nontrivial solution uof the
problem (such a solution is dened usually in a weak sense for a function ubelonging to a suitable
Sobolev-type space). Let us denote by  the set of all eigenvalues of problem (2.55) and dene
?:= inf :
Dene also the quotient
1:= inf
u2C1
0(
)nf0gZ

jrujp(x)dx
Z

jujp(x)dx:
Fan, Zhang & Zhao proved in [42] that ?0, the case ?= 0 can occur and ?>0, if and only if
1>0 (see, [42, Lemma 3.1]). Moreover, in [42, Theorem 3.2] has been showed that ifN= 1 then

2. Eigenvalue problems 41
1>0if and only if p(
)is a monotone function . Unfortunately, in high-dimensional spaces a necessary
and sucient condition relating the positivity of 1and the properties of function p(
) is not available
in the literature. However, some advances can be also found in [42, Theorem 3.1], where it was proved
that, if there is an open subset U
and a point x02Usuch thatp(x0)<(or>)p(x)for allx2@
,
then?= 0.On the other hand, by [42, Theorem 3.3] we know that ifN2and there exists a vector
l2RNnf0gsuch that for any x2
functionf(t) :=p(x+tl)is monotone for t2Ix:=ft:x+tl2
g
then1>0.In a similar context, Mih ailescu, R adulescu & Stancu-Dumitru showed that ifN2,
p2C1(
)and there exists a vectorial function !a:
!RNsuch that !a2C1(
;RN)and
9a0>0 s:t:div !a(x)a0>0;8×2
:
and
!a(x)
rp(x) = 0;8×2
;
then1>0 (see, [90, Theorem 1]). Note that similar results as those given by [90, Theorem 1] were also
obtained in a more general context by Allegretto in [2] and by Mih ailescu, Moro sanu & Stancu-Dumitru
in [87, Theorem 1].
The rst main result of this section is given by the following theorem.
Theorem 2.7. (see [47, Theorem 1.2]) Let !a:
!RNbe a vectorial function such that !a2
C1(
;RN)\C(
;RN). Assume that there exists a positive constant a0>0such that
div !a(x)a0>0;8×2
: (2.56)
Ifp2C1(
)\C(
)is a solution of the dierential inequality
!a(x)
rp(x)0;8×2
; (2.57)
then for each open set U
the minimum of pinUis achieved on @U.
Ifp2C1(
)\C(
)is a solution of the dierential inequality
!a(x)
rp(x)0;8×2
; (2.58)
then for each open set U
the maximum of pinUis achieved on @U.
Ifp2C1(
)\C(
)is a solution of the PDE's
!a(x)
rp(x) = 0;8×2
; (2.59)
then for each open set U
the maximum and minimum of pinUare achieved on @U.
Corollary 2.1. (see [47, Corollary 1.1]) Assume that condition (2.56) is satised. Assume p1,p22
C1(
)\C(
)are such that
!a(x)
rp1(x)(;=) !a(x)
rp2(x);8×2
:

2. Eigenvalue problems 42
LetU
be an open set, such that
p1(x)(;=)p2(x);8×2@U:
Then,
p1(x)(;=)p2(x);8x2U:
Remark 2. a. In the case when condition (2.59) in Theorem 2.7 is satised the conclusion of the
theorem remains true, if we replace assumption (2.56) by
div !a(x)a0<0;8×2
:
b. Note that in order to obtain the conclusion of Theorem 2.7, it is enough to have p2C1(
) (instead
ofp2C2(
) as in the case of Theorem 2.6).
c. Theorem 2.7 complements some earlier results obtained by Fan, Zhang & Zhao [42], Mih ailescu,
R adulescu & Stancu-Dumitru [90] and Mih ailescu, Moro sanu & Stancu-Dumitru [87, Proposition 1].
d. The analytic conclusion of Theorem 2.7 has a geometrical explanation. Indeed, from a geometrical
point of view the assumption (2.57) means that !aandrpform at every point of the domain an obtuse
angle which leads to the minimum of pon the boundary of the domain.
Example. We point out an example of functions pand !afor which Theorem 2.7 can be applied. Let
N= 2. Taking
= B1
31=3(0), !a(x) = (x1;2×2) andp(x) =x2
1×2+3
2it is easy to check that relations
(2.56) and (2.59) are fullled. Thus, pachieves its minimum and maximum on @
. Note also that
functionpgiven above is not harmonic since
p(x1;x2) = 2x2and consequently the above conclusion
can not be obtained by applying Theorem 2.6. More examples of functions satisfying conditions (2.56)
and (2.59) can be found in [90] and [87].
Next, we reformulate the maximum principle in Theorem 2.7 by relaxing condition (2.56). To be
more specic we prove
Theorem 2.8. (see [47, Theorem 1.3]) Let !a:
!RNbe a vectorial function, such that !a2
C1(
;RN)\C(
;RN). Assume
div !a(x) = 0;8×2
; (2.60)
and there exists a function A:
!(0;1), such that A2C1(
)\C(
)and for which there exists a
positive constant A0>0, such that
!a(x)
rA(x)A0>0;8×2
: (2.61)
Ifp2C1(
)\C(
)is a solution of the dierential inequality
!a(x)
rp(x)0;8×2
; (2.62)
then for each open set U
the minimum of pinUis achieved on @U.

2. Eigenvalue problems 43
Ifp2C1(
)\C(
)is a solution of the dierential inequality
!a(x)
rp(x)0;8×2
; (2.63)
then for each open set U
the maximum of pinUis achieved on @U.
Ifp2C1(
)\C(
)is a solution of the PDE's
!a(x)
rp(x) = 0;8×2
; (2.64)
then for each open set U
the maximum and minimum of pinUare achieved on @U.
Example. a. We point out an example of functions !a,pandAfor which Theorem 2.8 can be
applied. Indeed, it is enough to take N= 2,
f(x1;x2)2R2:9c >0;s:t: x1c; x 2cg,
!a(x1;x2) = (x1;x2),p(x1;x2) =x2
1×2
2andA(x1;x2) =x2
1×2+C, whereC > 0 is a constant suciently
large, such that A(x1;x2)>0, for each ( x1;x2)2
. Then conditions (2.60), (2.61) and (2.64) are
satised and consequently the maximum and minimum of pin
are achieved on @
.
b. On the other hand, we note that if in the above example we take
to be a ball centered in the
origin then min
p= 0 is achieved in (0 ;0), which does not belong to @
(actually, this represents an
example which shows that, if condition (2.56) is replaced by (2.60), then the conclusion of Theorem 2.7
does not hold true).
Proof of Theorem 2.7. First, we note that we can always assume that minx2
p(x)>1. Indeed,
it is easy to remark that if p(x) satises one of the relations (2.57), (2.58), (2.59) then C+p(x) satises
the corresponding relation too, for any real constant C. Thus, we may assume minx2
p(x)>1
eventually by choosing Cto be positive and large enough and replacing p(x) byC+p(x).
In the sequel we will use the following notations
p+:= max
x2
p(x); p:= min
x2
p(x):
Using the above remark we can assume that p>1.
We will prove Theorem 2.7 only in the case when conditions (2.56) and (2.57) are satised. The case
when conditions (2.56) and (2.58) hold true can be treated similarly in the sense that we can replace p(x)
byCp(x), whereCis a positive constant, large enough such that minx2
(Cp(x))>1. Thus,Cp(x)
will satisfy condition (2.57) and consequently we will infer that minx2U(Cp(x)) = minx2@U(Cp(x))
or maxx2Up(x) = maxx2@Up(x).
Finally, we note that in the case when conditions (2.56) and (2.59) are satised the conclusion of
Theorem 2.7 is obvious as a consequence of the above two cases.
In the following we prove Theorem 2.7, under assumptions (2.56) and (2.57). The conclusion of
Theorem 2.7 will follow as a consequence of two lemmas that will be presented below.

2. Eigenvalue problems 44
Denote
1:= inf
u2C1
0(
)nf0g;0u1Z

jrujp(x)dx
Z

jujp(x)dx:
Lemma 2.13. If there exists an open set U
and a point x02U, such that p(x0)< p(x)for all
x2@U, then1= 0.
Remark 3. The result of Lemma 2.13 represents a slight improvement of [42, Theorem 3.1]. The idea
of the proof below follows the lines of [42, Theorem 3.1].
Proof. Without loss of generality we may assume that U
and there exist two positive constants
0and1, such that
p(x0)<p(x)40;8×2@U; (2.65)
p(x0)<p(x)20;8x2B(@U; 1); (2.66)
where
B(@U; 1) :=fx2
:9y2@Us:t:jxyj<1g:
Indeed, since pis continuous on the compact set
it is clear that it is uniformly continuous.
Consequently, there exists 1>0 such that for each x,z2
withjxzj<1, we havejp(x)p(z)j<
20. It follows that, for each z2B(@U; 1), there exists xz2@U, such thatjzxzj<  1and
we havejp(xz)p(z)j<20. Then using (2.65) it is clear that for each z2B(@U; 1) we have
p(x0)<p(xz)40jp(xz)p(z)j+p(z)40<p(z)20. Thus, (2.66) is satised.
Next, we note that there exists a positive constant 2>0, such that
B(x0;2)UnB(@U; 1);
and
jp(x0)p(x)j<0;8x2B(x0;2): (2.67)
Obviously, there exists a function u2C1
0(
), such that
jru(x)jCand 0u(x)1;8×2
;
and
u(x) =8
><
>:0 for x62U[B(@U; 1)
1 for x2UnB(@U; 1):

2. Eigenvalue problems 45
Next, it is clear that for each t2(0;1), we have
1Z

jr(tu)jp(x)dx
Z

jtujp(x)dxZ
B(@U; 1)jr(tu)jp(x)dx
Z
B(x0;2)jtujp(x)dx
Z
B(@U; 1)jtCjp(x)dx
Z
B(x0;2)jtjp(x)dxC1jtCjp(1)
C2jtjp(2);
whereC1:=jB(@U; 1)j,C2:=jB(x0;2)j,12B(@U; 1) and22B(x0;2).
Using relations (2.66) and (2.67), we nd
p(1)p(2)>0;
and we get 1C3t0, for allt2(0;1) and therefore 1= 0.
The proof of Lemma 2.13 is complete. 
Lemma 2.14. Assume that conditions (2.56) and (2.57) are fullled. Then, 1>0.
Remark 4. The proof of Lemma 2.14 will follow the lines of [90, Theorem 1] and is inspired by the
ideas in [106, Th eor eme 20.7].
Proof. Simple computations show that for each u2C1
0(
) with 0u1, the following equality
holds true
div(u(x)p(x) !a(x)) =u(x)p(x)div !a(x) +p(x)u(x)p(x)1ru(x)
!a(x) +
u(x)p(x)log(u(x))rp(x)
!a(x):
On the other hand, the
ux-divergence theorem implies that for each u2C1
0(
) with 0u1, we
have Z

div(u(x)p(x) !a(x))dx=Z
@
u(x)p(x) !a(x)
!n d (x) = 0:
Using the above pieces of information and condition (2.57), we infer that for each u2C1
0(
) with
0u1 it holds true that
Z

u(x)p(x)div !a(x)dx=Z

p(x)u(x)p(x)1ru(x)
!a(x)dx
Z

u(x)p(x)log(u(x))rp(x)
!a(x)
p+Z

u(x)p(x)1jru(x)jj !a(x)jdx:
Next, we recall that for each >0, eachx2
and each A,B0, the following Young type inequality
holds true (see, e.g. [21, the footnote on p. 56])
ABAp(x)
p(x)1+1
p(x)1Bp(x):

2. Eigenvalue problems 46
We x>0, such that
p+<a 0;
wherea0is given by relation (2.56).
The above facts and relation (2.56) yield
a0Z

u(x)p(x)dxp+"
Z

u(x)p(x)dx+Z

1
p(x)1
j !a(x)jp(x)jru(x)jp(x)dx#
;
for anyu2C1
0(
) with 0u1, or
(a0p+)Z

u(x)p(x)dx"1
p1
+1
p+1#
p+Z

j !a(x)jp(x)jru(x)jp(x)dx;
for anyu2C1
0(
) with 0u1.
The proof of Lemma 2.14 is complete. 
Proof of Theorem 2.8.
The same arguments as in the proof of Theorem 2.7 can be used in order to consider that p>1.
On the other hand, similar arguments as in the proof of Theorem 2.7 assure that it is enough to give the
proof of Theorem 2.8 in the case when conditions (2.60), (2.61) and (2.62) are satised. Undoubtedly,
Lemma 2.13 still holds true in the hypothesis of Theorem 2.8. We prove the following auxiliary result:
Lemma 2.15. Assume that conditions (2.60), (2.61) and (2.62) are fullled. Then, 1>0.
Proof. By relation (2.60), it follows that
Z

!a(x)
r'(x)dx= 0;8'2C1
0(
):
Next, note that for each u2C1
0(
) with 0u1, we have '=u(x)p(x)A(x)2C1
0(
) and since
r'(x) =u(x)p(x)rA(x) +A(x)u(x)p(x)log(u(x))rp(x) +A(x)u(x)p(x)1p(x)ru(x);
we get by using relation (2.62) that
Z

u(x)p(x)rA(x)
!a(x)dx=Z

A(x)u(x)p(x)1p(x)ru(x)
!a(x)dx
Z

A(x)u(x)p(x)log(u(x))rp(x)
!a(x)dx;
for anyu2C1
0(
) with 0u1.
The proof continues as in the case of Lemma 2.14. 
Consequently, Theorem 2.8 holds true. 

Chapter 3
Torsional creep problems
In a general context the creep suggests the inelastic response of materials at high temperatures. If
we refer to a prismatic material rod, its torsional creep represents the plastic deformation of the rod,
subject to a torsional moment for an extended period of time at high temperature. Let
RN(N2)
be a bounded domain with smooth boundary @
. In the particular case when N= 2,
can be regarded
as the cross section of the rod. According to Kachanov [70, 71], for each real number p>1, torsional
creep problems are modeled by the non-homogeneous equations
8
><
>:div(jrujp2ru) = 1 in
u= 0 on @
;(3.1)
which possess unique solutions denoted by up. The limit problem of the above family of equations, as
p!1 , is given by 8
><
>:minfjruj1;
1ug= 0 in
u= 0 on @
;(3.2)
and it possesses as unique (viscosity) solution (see Jensen [65] and Juutinen [67]) the distance function
to the boundary of
, i.e. :
![0;1), dened by (x) := infy2@
jxyj,x2
. In connection with
the discussion concerning the torsional creep problems, the limiting problem (3.2) models the perfect
plastic torsion . In [16], Bhattacharya, DiBenedetto & Manfredi proved that fupgconverges uniformly
toin
. Independently and simultaneously, the uniform convergence was also obtained by Kawohl in
[72]. Historically, this result follows the rst advance in this direction obtained by Payne & Philippin
in [92], where they established the convergenceR

updx!R

dx, asp!1 .
More recently, Perez-Llanos & Rossi [93] and Bocea & Mih ailescu [19] generalized the above results
by showing that they continue to apply if the p-homogeneous dierential operator div( jrujp2ru),
arising in equation (3.1) is replaced by dierent classes of inhomogeneous dierential operators. To be
47

3. Torsional creep problems 48
more precise, in [93] was considered the family of equations
8
><
>:div(jrujpn(x)2ru) = 1 in
u= 0 on @
;(3.3)
wherepn:
!(1;1) is a sequence of suciently smooth functions, such that pn(x) converges
uniformly to innity in
and the limits
lim
n!1rlnpn(x) =(x) & lim sup
n!1max
pn
min
pnk;for somek>0;
exist, while in [19] was investigated the case of problems
8
><
>:div
'n(jruj)
jrujru
='n(1) in
u= 0 on @
;(3.4)
where, for each integer n>1, the mappings 'n:R!Rare odd, increasing homeomorphisms of class
C1satisfying
0<'
n1t'0
n(t)
'n(t)'+
n1<1;8t0; (3.5)
for some constants '
nand'+
nwith 1<'
n'+
n<1;
'
n!1 asn!1; (3.6)
and such that
there exists a real constant  >1;such that'+
n'
n;for alln>1: (3.7)
Note that the results from [93] and [19] are indeed generalizations of the classical results obtained in the
case of problem (3.1), since taking pn(x) =nin (3.3) or'n(t) =njtjn2tin (3.4), we recover problem
(3.1) withp=n.
Motivated by the above advances, in the following, the goal of this chapter is to show that similar
results to those obtained by Kawohl in the case of problem (3.1) continues to hold true under more
general constitutive assumptions.
3.1 On the convergence of the sequence of solutions for a family of
eigenvalue problems
In this section we consider the family of eigenvalue problems
8
>>>><
>>>>:
2nu=u; forx2
u= 0; forx2@
jjujjL2(
)= 1;(3.8)

3. Torsional creep problems 49
where
2nu:= div(jruj2n2ru) stands for the 2 n-Laplace operator and is a real number.
We say that 2Ris an eigenvalue of problem (3.8), if there exists un2W1;2n
0(
)nf0g(with
kunkL2(
)= 1), such thatZ

jrunj2n2runrvdx =Z

unvdx; (3.9)
for allv2W1;2n
0(
). The function unfrom the above denition will be called an eigenfunction
corresponding to the eigenvalue . Our goal here will be to show that in the case when is the
lowest eigenvalue of problem (3.8), the sequence unconverges uniformly to a certain limit that will be
identied. More precisely, our main result will be given by the following theorem.
Theorem 3.1. (see [50, Theorem 1]) For each integer n1dene
1(n) := inf
u2W1;2n
0(
)nf0gZ

jruj2ndx
Z

u2dxn:
Then1(n)is a positive real number, which gives the lowest eigenvalue of problem (3.8). Letting unbe
a corresponding positive eigenfunction, the sequence fungconverges uniformly in
tokk1
L2(
).
Our strategy to prove Theorem 3.1 will be the following. First, we will check that for each pos-
itive integer n, the quantity 1(n) is the lowest eigenvalue of problem (3.8) with the corresponding
eigenfunction a positive minimizer of 1(n), sayun. Next, using similar ideas as those developed by
Juutinen, Lindqvist & Manfredi in [68], we will show that unconverges uniformly in
to a limiting
function, say u1. We will identify the limiting equation which has as a viscosity solution function u1.
Finally, using a maximum principle introduced by Jensen in [65], we will show that u1=kk1
L2(
).
We start by pointing out a few remarks on 1(n).
Lemma 3.1. For each integer n1, we have1(n)>0.
Proof. Note that by H older's inequality, we have
Z

u2dxj
j(n1)=nZ

u2ndx1=n
;8u2L2n(
):
On the other hand, it is well-known that letting
1(2n) := inf
u2W1;2n
0(
)nf0gZ

jruj2ndx
Z

juj2ndx;

3. Torsional creep problems 50
this is a positive real number (the rst eigenvalue of the 2 n-Laplace operator with homogeneous Dirichlet
boundary condition). Combining the above pieces of information, we get
0<1(2n)
j
jn1Z

jruj2ndx
Z

u2dxn;8u2W1;2n
0(
);
and the conclusion of Lemma 3.1 follows. 
Lemma 3.2. For each integer n1, there exists un2W1;2n
0(
), such that un>0in
,R

u2
ndx= 1
andR

jrunj2ndx=1(n). Moreover, 1(n)is an eigenvalue of problem (3.8) and unis the corre-
sponding eigenfunction.
Proof. Letfvmgm12W1;2n
0(
)nf0gbe a minimizing sequence for 1(n), that is
Z

jrvmj2ndx
Z

v2
mdxn!1(n) asm!1:
Denewm(x) :=vm(x)R

v2mdx. Then, we havekwmkL2(
)= 1, for each integer mand
lim
m!1Z

jrwmj2ndx=1(n):
It follows thatfwmgmis bounded in W1;2n
0(
), which implies that there exists un2W1;2n
0(
) such
thatwmconverges weakly to uninW1;2n
0(
). Since W1;2n
0(
) is compactly embedded in L2(
), we
deduce that wmconverges strongly to uninL2(
) and, thus,kunkL2(
)= 1. Since, by [21, Proposition
III.5 (iii)], we haveZ

jrunj2ndxlim inf
m!1Z

jrwmj2ndx;
we deduce by the denition of 1(n) that
1(n) =Z

jrunj2ndx:
In other words, we proved the existence of a minimizer for 1(n) satisfyingkunkL2(
)= 1. Sincejunj
will be a minimizer for 1(n), too, we may assume that un0 in
.
To go further, let v2W1;2n
0(
) be arbitrary but xed. We dene the function f:R!Rby
f(t) :=Z

jr(un+tv)j2ndx1(n)Z

jun+tvj2dxn
:
It is clear that f2C1(R) andt= 0 is a minimum point for f. It follows that f0(0) = 0, which implies
that Z

jrunj2n2runrvdx =1(n)Z

unvdx:

3. Torsional creep problems 51
Sincevwas xed arbitrary, we nd that 1(n) is an eigenvalue of (3.8) and unis a nonnegative corre-
sponding eigenfunction. Finally, the positivity of unfollows from Harnack's inequality (see, Trudinger
[104, Theorem 1.1, p.724 and Corollary 1.1, p. 725] or L^ e [75, Theorem 4.6] for the statement of Har-
nack's inequality and L^ e [75, Lemma 5.3] for details regarding the application of Harnack's inequality
to obtain the positivity of unin
). 
Lemma 3.3.
lim
n!1np
1(n) =1
kk2
L2(
):
Proof. We prove this lemma in 3 steps.
Step 1. Dene the linear space
X0:=W1;1(
)\
\q1W1;2q
0(
)
:
We show that
inf
u2X0nf0gkjruj2kL1(
)
kuk2
L2(
)=1
kk2
L2(
): (3.10)
First, letu:
!Rbe a Lipschitz continuous function with u= 0, on@
. For each x2
, there exists
y2@
such that (x) =jxyj. Letg: [0;1]!R;
g(t) :=u(tx+ (1t)y);8t2[0;1]:
It is clear that g2C([0;1])\C1(0;1). By the mean value theorem, we deduce that there exists
tx2(0;1) such that
g0(tx) =g(1)g(0);
or,
ju(x)j=ju(x)u(y)j=jg(1)g(0)j=jru(txx+ (1tx)y)
(xy)j;
and thus, via the Cauchy-Schwarz inequality, we deduce
ju(x)j2jxyj2jru(txx+ (1tx)y)j2(x)2sup
z2[x;y]jru(z)j2=(x)2kjruj2kL1(
):
Since the last inequality holds true for each x2
, we can integrate over
in order to nd that
kuk2
L2(
)=Z

ju(x)u(y)j2dxkjruj2kL1(
)
kk2
L2(
):
We deduce that
1
kk2
L2(
)kjruj2kL1(
)
kuk2
L2(
); (3.11)
for each Lipschitz continuous function u:
!Rwhich vanishes on @
.
Next, letu2X0. Thenu2W1;q
0(
), for each q>1, andu2W1;1(
). Applying Morrey's theorem
(see, e.g. [21, Theorem 9.12]), we have
ju(x)u(y)jC(q;N)jxyj1N
qkjrujkL2q(
);8q>N and a.e.x;y2
;

3. Torsional creep problems 52
whereC(q;N) is a positive constant. Moreover, by [6, Theorem 5.7.4] we deduce that we can take
C(q;N) :=22N=q
1N=q. Lettingq!1 in the above inequality, we get
ju(x)u(y)j4jxyjkjrujkL1(
);a.e.x;y2
;
from which, we deduce that uis a Lipschitz continuous function. On the other hand, since u2
\q1W1;q
0(
), thenu2W1;N+1
0 (
) and since W1;N+1
0 (
) is compactly embedded in C(
), by [21,
Theorem 9.17], we deduce that u= 0 on@
. In other words, each function u2X0is a Lipschitz
continuous function which vanishes on @
. Thus we conclude that relation (3.11) holds true for each
u2X0nf0g. Finally, recall that is a Lipschitz continuous function which vanishes on @
and satises
jr(x)j= 1;a:e:x2
:
In addition, since 2W1;q
0(
), for all q1 and2W1;1(
), by the denition of X0, we have that
2X0nf0g. From the above pieces of information we infer that relation (3.10) holds true.
Step 2. We show that
lim sup
n!1np
1(n)1
kk2
L2(
): (3.12)
By the denition of 1(n) and the fact that 2W1;2n
0(
)nf0g, we have that
np
1(n)Z

jrj2ndx1=n
Z

2dx=kjrj2kLn(
)
kk2
L2(
);8n>1:
Lettingn!1 and taking into account that jrj= 1 a.e. in
relation (3.12) follows.
Step 3. We check that
lim inf
n!1np
1(n)1
kk2
L2(
): (3.13)
For each integer n1, letun2W1;2n
0(
) be given by Lemma 3.2. Thus, in particular, we have thatR

u2
ndx= 1 andR

jrunj2ndx=1(n). By Step 2, we have
np
1(n)1
kk2
L2(
)+ 1;fornlarge enough :
For each real number m > N and each integer n > m , applying H older's inequality and taking into
account the above pieces of information we have
Z

jrunj2mdxZ

jrunj2ndxm
n

j
j1m
n= [1(n)]m
n
j
j1m
n; (3.14)
or
kjrunj2kLm(
)np
1(n)
j
j1
m1
n
1
kk2
L2(
)+ 1!
(j
j+ 1): (3.15)

3. Torsional creep problems 53
We obtain that the sequence fungn>m is bounded in W1;2m
0(
). It follows that, there exists a subse-
quence offung, still denoted by fung, andu12W1;2m
0(
), such that unconverges weakly to u1in
W1;2m
0(
) andun!u1inL2(
). Thus,ku1kL2(
)= 1 and, consequently, u16= 0. We also note
that, sincem>N , the compactness of the embedding of W1;2m
0(
) inC(
) allows us to conclude that
unconverges uniformly to u1in
andu12C(
). Sinceunconverges weakly to u1inW1;2m
0(
)
and taking into account (3.15), we have
Z

jru1j2mdx1
m
lim inf
n!1Z

jrunj2mdx1
m

1
kk2
L2(
)+ 1!
(j
j+ 1);8m>N: (3.16)
Further, we show that u12W1;1(
). Indeed, since u12C(
), thenu12L1(
). It remains just
to prove thatjru1j2L1(
). To this end, let x2
be a Lebesgue point1forjru1j2L1(
) and
letr2(0;(x)). Sinceunconverges weakly to u1inW1;2m
0(
), thenunconverges weakly to u1in
W1;1
0(
) and for any ball Br(x)
, we have
Z
Br(x)jru1jdylim inf
n!1Z
Br(x)jrunjdy: (3.17)
By H older's inequality, we have
Z
Br(x)jrunjdy Z
Br(x)jrunj2mdy!1
2m Z
Br(x)1dy!11
2m
=kjrunj2k1=2
Lm(Br(x))
jBr(x)j11
2m;
or1
jBr(x)jZ
Br(x)jrunjdy1
jBr(x)j1
2mkjrunj2k1=2
Lm(Br(x)): (3.18)
By relations (3.15), (3.17) and (3.18), we obtain
1
jBr(x)jZ
Br(x)jru1jdylim inf
n!11
jBr(x)j1
2mkjrunj2k1=2
Lm(Br(x))
1
jBr(x)j1
2m
1
kk2
L2(
)+ 1!1=2
(j
j+ 1)1=2;8m>N:
Lettingm!1 in the above relation, we get
1
jBr(x)jZ
Br(x)jru1jdy
1
kk2
L2(
)+ 1!1=2
(j
j+ 1)1=2;8r2(0;(x)):
1Ifw:
!Ris Lebesgue integrable then x2
is a Lebesgue point of wif
lim
r!0+1
jBr(x)jZ
Br(x)jw(y)w(x)jdy= 0:
Ifw2L1(
) then almost every x2
is a Lebesgue point of w.

3. Torsional creep problems 54
Since almost every x2
is a Lebesgue point for jru1j, passing to the limit r!0+in the above
inequality, we get
jru1(x)j
1
kk2
L2(
)+ 1!1=2
(j
j+ 1)1=2;for a.e.x2
;
which implies that jru1j2L1(
). Thus, we showed that u12W1;1(
).
Next, since unconverges weakly to u1inW1;2m
0(
) and relations (3.16) and (3.14) hold true we
deduce
kjru1j2kLm(
)
ku1k2
L2(
)lim inf
n!1kjrunj2kLm(
)
ku1k2
L2(
)
lim inf
n!1np
1(n)
j
j1
m1
n
ku1k2
L2(
)
=j
j1
m
ku1k2
L2(
)lim inf
n!1np
1(n);8m>N:
Taking into account that u12W1;1(
) and letting m!1 in the above inequality, we get
kjru1j2kL1(
)
ku1k2
L2(
)1
ku1k2
L2(
)lim inf
n!1np
1(n): (3.19)
On the other hand, by the facts that u12W1;1(
) andu12W1;2m
0(
), for each m>N , we deduce,
via [21, Theorem 9.17] that u1= 0 on@
and, consequently, u12X0nf0g. Then, by Step 1 we nd
1
kk2
L2(
)kjru1j2kL1(
)
ku1k2
L2(
): (3.20)
Combining relations (3.19) and (3.20), we get (3.13).
By (3.12) and (3.13), we obtain the conclusion of Lemma 3.3. 
Lemma 3.4. Letfungbe the sequence given by Lemma 3.2. There exists a subsequence of fung, still
denoted byfung, which converges uniformly in
, asn!1 to a function u12C(
)nf0g.
Proof. This is established in the proof of Step 3 from Lemma 3.3. 
Assuming that solution unof equation (3.8) is smooth enough, we can rewrite equation (3.8) as
jrunj2n2
un(2n2)jrunj2n4
1un=1(n)un; (3.21)
where
1
:=PN
i;j=1@

@xi@

@xj@2

@xi@xjis the so-called1-Laplace operator. This equation is nonlinear, but
elliptic, thus it makes sense to consider its viscosity solutions.
Lety2R,z2RnandSbe a real symmetric matrix in MNN. Consider the following continuous
function
Hn(y;z;S ) =jzj2n2Trace(S)(2n2)jzj2n4hSz;zi1(n)y:

3. Torsional creep problems 55
We are interested in nding viscosity solutions of the partial dierential equation
8
><
>:Hn(u;ru;D2u) = 0;forx2
u= 0; forx2@
:(3.22)
Denition 3.1. An upper semicontinuous function u:
!Ris a viscosity subsolution of (3.22) if
uj@
0and, whenever x02
and 2C2(
)are such that u(x0) = (x0)andu(x)< (x), for all
x2Br(x0)nfx0g, then
Hn( (x0);r (x0);D2( (x0)))0:
Denition 3.2. A lower semicontinuous function u:
!Ris a viscosity supersolution of (3.22) if
uj@
0and, whenever x02
and 2C2(
)are such that u(x0) = (x0)andu(x)> (x), for all
x2Br(x0)nfx0g, then
Hn( (x0);r (x0);D2( (x0)))0:
Denition 3.3. A continuous function u:
!Ris a viscosity solution of (3.22) if it is both a viscosity
subsolution and a viscosity supersolution of (3.22).
The following result can be obtained by using the ideas used by Juutinen, Lindqvist & Manfredi in
[68, Lemma 1.8]. We include the proof for readers' convenience.
Lemma 3.5. A continuous weak solution of (3.8) is a viscosity solution of (3.22).
Proof. We start by checking that unis a viscosity supersolution of (3.22). Let x02
and let
2C2(
) be a test function such that un(x0) = (x0) andun(x)> (x), for allx2Br(x0)nfx0g.
We want to show that

2n (x0) =jr (x0)j2n2
(x0)(2n2)jr (x0)j2n4
1 (x0)1(n) (x0):
Assume by contradiction that this is not the case. By continuity, there exists a radius r>0, such that
jr (x)j2n2
(x)(2n2)jr (x)j2n4
1 (x)< 1(n) (x);
for everyx2Br(x0)nfx0g. Takingrsmaller if necessary, we may assume that un> inBr(x0)nfx0g.
Setm:= infjxx0j=r(un )(x) and dene w(x) := (x) +m
2. The function wveriesw(x0)>un(x0)
andw(x)<un(x), for allx2@Br(x0). Moreover, it holds that

2nw(x)< 1(n) (x);inBr(x0): (3.23)
Multiplying (3.23) by ( wun)+, which vanishes on @Br(x0), we get
Z
Br(x0)\[w>un]jrw(x)j2n2rw(x)r(wun)(x)dx< 1(n)Z
Br(x0)\[w>un] (x)(wun)(x)dx:

3. Torsional creep problems 56
Testing in (3.9) with ( wun)+extended by zero outside Br(x0) we obtain
Z
Br(x0)\[w>un]jrun(x)j2n2run(x)r(wun)(x)dx=1(n)Z
Br(x0)\[w>un]un(x)(wun)(x)dx:
Subtracting the last two relations and using the fact that un> onBr(x0)nfx0g, we get
0Z
Br(x0)\[w>un](jrw(x)j2n1jrun(x)j2n1)(jrw(x)jjrun(x)j)dx
Z
Br(x0)\[w>un](jrw(x)j2n2rw(x)jrun(x)j2n2run(x))r(w(x)un(x))dx
<  1(n)Z
Br(x0)\[w>un][ (x)un(x)](w(x)un(x))dx0;
a contradiction. Therefore, unis a viscosity supersolution of problem (3.22). The fact that unis a
viscosity subsolution runs as above and we omit the details. 
By Lemma 3.4, we may extract a subsequence un!u1uniformly in
asn!1 . Next, our goal
is to identify the limit equation veried by u1. To do that, for z2RnandSa real symmetric matrix,
we dene the limit operator H1by
H1(z;S) := minfhSz;zi;jzjjjjj1
L2(
)g:
Theorem 3.2. Letu1be the function obtained as a uniform limit of a subsequence of fungin Lemma
3.4. Then, u1is a viscosity solution of problem
8
><
>:H1(ru;D2u) = 0;forx2
u= 0; forx2@
:(3.24)
Proof. First, we show that u1is a supersolution of (3.24). Fix x02
and a function 2C2(
),
such thatu1(x0) = (x0) andu1(x)> (x), for anyx2Br(x0)nfx0g. Sinceun!u1uniformly,
there exists a sequence fxngn
such that xn!x0,un(xn) = (xn) andun has a local minimum
atxn(see details for example in Bocea, Mih ailescu & Stancu-Dumitru [20, Theorem 3.1]).
By Lemma 3.5, the function unis a viscosity solution of (3.22) and therefore
jr (xn)j2n2
(xn)(2n2)jr (xn)j2n4
1 (xn)1(n) (xn): (3.25)
Lemma 3.2 implies that un>0 in
. Then (xn) =un(xn)>0 for eachn, which in view of (3.25)
implies thatjr (xn)j6= 0. Multiplying (3.25) by 1 =[(2n2)jr (xn)j2n4], we get
jr (xn)j2
2n2
(xn)
1 (xn)1(n) (xn)
(2n2)jr (xn)j2n4: (3.26)
Lettingn!1 , we obtain

1 (x0)lim sup
n!1"
1(n)1=(2n) (xn)1=(2n)
(2n2)1=(2n)jr (xn)j12=n#2n
:

3. Torsional creep problems 57
In particular, we nd

1 (x0)0: (3.27)
By Lindqvist & Manfredi [82, Denitions 2.1 and 2.2 and Proposition 2.3], we deduce that u1is an
1-superharmonic function. Recalling that u1is also nonnegative, it follows by the Harnack-type
inequality [82, Corollary 4.5] that u1is actually positive in
. In particular, this information yields
that (x0) =u1(x0)>0.
Next, we claim that
jr (x0)jjjjj1
L2(
)0: (3.28)
Suppose that this is not the case, and then
jjjj1
L2(
)
jr (x0)j>1:
Using that fact and Lemma 3.3, we deduce
lim
n!11(n)1=(2n) (xn)1=(2n)
(2n2)1=(2n)jr (xn)j12=n=jjjj1
L2(
)
jr (x0)j>1:
It follows that there exists 0>0 such that
1(n)1=(2n) (xn)1=(2n)
(2n2)1=(2n)jr (xn)j12=n1 +0;for eachn2Nsuciently large :
The above estimates imply
lim sup
n!1"
1(n)1=(2n) (xn)1=(2n)
(2n2)1=(2n)jr (xn)j12=n#2n
lim
n!1(1 +0)2n=1;
which contradicts (3.27). Thus, (3.28) holds true.
Therefore (3.27) and (3.28) yield
minf
1 (x0);jr (x0)jjjjj1
L2(
)g0: (3.29)
To end the proof of this theorem, it remains to check that u1is a viscosity subsolution of (3.24).
Letx02
be xed and let 2C2(
) be a test function such that u1(x0) = (x0) andu1(x)< (x),
for anyxin a neighborhood of x0. We have to check that
minf
1 (x0);jr (x0)jjjjj1
L2(
)g0:
Ifjr (x0)j= 0, then the above inequality obviously holds true. It suces to show that if jr (x0)j>0
and
jr (x0)jjjjj1
L2(
)>0; (3.30)

3. Torsional creep problems 58
then
1 (x0)0. We follow the arguments considered in the supersolution case and we can
construct a sequence xn!x0asn!1 , such that
jr (xn)j2
2n2
(xn)
1 (xn)"
1(n)1=(2n) (xn)1=(2n)
(2n2)1=(2n)jr (xn)j12=n#2n
:
Lettingn!1 from (3.30), we obtain

1 (x0)lim inf
n!1"
1(n)1=(2n) (xn)1=(2n)
(2n2)1=(2n)jr (xn)j12=n#2n
= 0;
which ends the proof. 
Proof of Theorem 3.1. By Theorem 3.2, we deduce that there exists a function u1, which is
the uniform limit of a subsequence of fungand solves in the viscosity sense the problem
8
><
>:minf
1u;jrujkk1
L2(
)g= 0;forx2
u= 0; forx2@
:(3.31)
On the other hand, it is obvious that u=kk1
L2(
)is a solution of the above problem, too. But, using
Jensen's maximum principle [65, Theorem 2.1] (see also Ishibashi & Koike [64, p. 546]), we deduce the
uniqueness of the solution of equation (3.31), which also implies the fact that the entire sequence un
converges uniformly in
to kk1
L2(
). The proof of Theorem 3.1 is complete. 
3.2 On a family of torsional creep problems involving rapidly growing
operators in divergence form
In this section, our goal is to investigate problem (3.4) in the case when, for each integer n>1, we take
'n(t) :=pnjtjpn2tejtjpn;8t2R; (3.32)
wherepn2(1;1) are given real numbers such that lim n!1pn= +1. Note that this case is not
covered by the study from [19], since simple computations show that for each integer n >1, we have
supt>0t'0
n(t)
'n(t)= +1, if'nis given by (3.32), and, thus, there does not exist any constant '+
n2(1;1),
for which condition (3.5) holds true. As we will see in the following, this new case also possesses other
diculties related to the properties of the function spaces where the problem is analyzed, with the
denition of a variational solution for problem (3.4) in the new context and with the analysis of the
\passage to the limit ", which requires a more careful treatment.
We recall that equations involving operators built with the aid of functions 'n, given in relation
(3.32), are known in the literature as problems involving rapidly growing dierential operators . The
study of equations for rapidly growing operators goes back several decades (see, e.g., Donaldson [41],
Gossez [61, 62], Lieberman [77], Le & Schmitt [76]). More recently, some eigenvalue problems for
dierential operators of exponential type have been studied in the framework of Orlicz-Sobolev spaces
(see, e.g., Garc a-Huidobro et. al [57], Bocea & Mih ailescu [17]).

3. Torsional creep problems 59
3.2.1 Function spaces
Throughout this section, we x some integer n > 1 arbitrary. Let us construct the natural function
space to analyze problem (3.4) with the choice 'nspecied in (3.32). We start by introducing an
Orlicz-Sobolev space that will be constructed according to [17, Section 2], where the case pn= 2 was
emphasized. Let  n:R!Rbe the antiderivative of 'n, given by  n(t) :=Rt
0'n(s)ds=ejtjpn1.
It is easy to check that 'nis an odd, increasing homeomorphism and, thus,  nis anN-function (see,
e.g., [1, Chapter VIII]). The complementary N-function of  nis dened by 
n(t) :=Rt
0'1
n(s)ds. The
explicit expression of 
nis not trivial to compute; however one can show that it is itself an N-function
(see, e.g., [98, Chapter II, p. 15]). It can be easily established that
lim
t!0t'n(t)
n(t)=pnand lim
t!1t'n(t)
n(t)=1:
Since the map (0 ;1)3t!t'n(t)
n(t)is increasing, we deduce that
pnt'n(t)
n(t);8t>0: (3.33)
Using (3.33), in view of [55, Lemma 2.1], we have
n(1)tpnn(t);8t2(1;1): (3.34)
Dene the Orlicz class
Kn(
) :=
u:
!Rmeasurable:Z

n(ju(x)j)dx<1
;
and letLn(
) be the Orlicz space associated to  n, that is, the linear hull of Kn(
). The space
Ln(
) endowed with the Luxemburg norm
kukn:= inf
>0 :Z

nju(x)j

dx1
;
is a Banach space. Let En(
) be the closure of L1(
) inLn(
) with respect to k
k n. It can be
shown that En(
)Kn(
)Ln(
). Since  ndoes not satisfy the
2-condition at innity (see,
e.g., [1] for the denition of this condition), En(
) is a proper closed subspace of Ln(
). However,
its complementary, 
n, satises the
2-condition (see, e.g., [73, p. 28] for details). It is also known
thatEn(
) is the maximal linear subspace of Kn(
) (see, e.g., [1, Lemma 8.15]) and it is separable
(see, e.g., [1, Theorem 8.21(b)]). Moreover, Ln(
) is the dual space of E
n(
), whileL
n(
) is the
dual space of En(
) [57, Theorem 2.1]. Recall also the following useful result.
Lemma 3.6. For each real number q1, we have the continuous embedding Ln(
)Lq(
).
Proof. For 1q <1, we consider the function q(t) :=jtjq
qand since lim t!1 q(kt)
n(t)= 0, for
allk >0, we have that  ndominates qnear innity. Applying [1, Lemma 8.12(b)], we obtain the
conclusion of the lemma. 

3. Torsional creep problems 60
The corresponding Sobolev space of functions in Ln(
) (resp. En(
)) with distributional rst
derivatives belonging to Ln(
) (resp. En(
)) is denoted by W1;n(
) (andW1En(
), respec-
tively).W1;n(
) is a Banach space when endowed with the norm kuk1;n:=kukn+kjrujkn.
Note that the spaces W1;n(
) (resp. W1En(
)) can be identied with a closed subspace of the
product N
i=0Ln(
) (and N
i=0En(
), respectively). According to the above pieces of information
we have N
i=0Ln(
) =
N
i=0E?
n(
)
?and if we denote by :=
N
i=0Ln(
);N
i=0E?
n(
)

the
weak?topology in N
i=0Ln(
) and also the restriction of to the closed subspace W1;n(
), then
W1;n(
) is closed under weak?convergence of N
i=0Ln(
). In particular, that fact means that, if
fukgis a bounded sequence in W1;n(
) (with respect to k
k1;n), thenfukghas a subsequence which
converges to the weak?topologyto someu2W1;n(
), i.e. a bounded set in W1;n(
) is relatively
(sequentially) compact with respect to weak?topology.
Next, in order to obtain an adequate function space where to analyze problem (3.4), we have to
take into account the boundary condition. For that reason, we dene the linear space
Xn:=W1;n(
)\
\q>1W1;q
0(
)
:
It can be shown that ( Xn;k
k 1;n) is a closed subspace of ( W1;n(
);k
k 1;n) and that, iffukgXn
is a bounded sequence in W1;n(
) (that means with respect to the norm k
k1;n), thenfukgcontains
a subsequence which converges in the sense of the weak?topologyto someu2Xn. Let us provide
the proofs of these results.
Proposition 3.1. (Xn;k
k 1;n)is a closed subspace of (W1;n(
);k
k 1;n).
Proof. LetfukgXnbe a sequence, which converges to u2W1;n(
) ink
k 1;n. By Lemma 3.6,
we know that W1;n(
)W1;q(
), for all q>1. It means that ukalso converges to uin the norm of
k
kW1;q(
). In particular, since fukgXnW1;q
0(
), for all q >1, we deduce that u2W1;q
0(
). It
follows that u2Xn.
Proposition 3.2. IffukgXnis a bounded sequence with respect to the norm k
k 1;nthenfukg
contains a subsequence which converges in the sense of the weak?topologyto someu2Xn.
Proof. Indeed, it is clear that there exists u2W1;n(
) such thatfukghas a subsequence which
converges to the weak?topologytou. SinceW1;n(
) is compactly embedded in L1(
), we deduce
that, by passing to a subsequence, if necessary, then ukconverges strongly to uinL1(
) (see [76,
Lemma 5] for details) and next, by passing again to a subsequence, uk(x) converges to u(x) for a.e.
x2
. On the other hand, since fukgXn, it follows thatfukgW1;q
0(
), for all q >1. Fixq >1.
The fact that W1;n(
)W1;q(
) assures thatfukgis bounded in W1;q(
). It means that there exists
uq2W1;q
0(
) such that, passing eventually to a subsequence, ukconverges weakly to uqinW1;q
0(
).
The compact embedding of W1;q
0(
) intoL1(
) implies that ukconverges strongly to uqinL1(
) and
then it possesses a subsequence, still denoted by uk, such that uk(x) converges to uq(x) for a.e.x2
.
The above pieces of information lead to the fact that uq=ua.e. in
and consequently, u2W1;q
0(
).
In this manner we found that u2Xn.

3. Torsional creep problems 61
3.2.2 Variational solutions
The Euler-Lagrange functional associated to problem (3.4) is In:Xn!R, dened by
In(u) :=1
'n(1)Z

n(jruj)dxZ

udx:
IfInwas smooth on Xn, then one could dene a (weak) solution of problem (3.4), as a function un2Xn,
such that1
'n(1)Z

'n(jrunj)
jrunjrunrvdxZ

vdx = 0;8v2Xn:
This is the common denition of a weak solution used in the study of equation (3.4) in [19] under
assumptions (3.5), (3.6) and (3.7).
Unfortunately, in the new framework, the functional Inis not smooth on Xn. Indeed, even if the
functionalgn:Xn!R, dened by
gn(u) :=Z

udx;
belongs toC1(Xn;R) (see, [57, Lemma 3.5]), and we have
hg0
n(u);vi=Z

vdx;8u; v2Xn;
the functional hn:Xn!R, given by
hn(u) :=1
'n(1)Z

n(jruj)dx;
does not belong to C1(Xn;R). However, hnpossesses some remarkable properties, namely, it is convex,
weaklylower semicontinuous (see, [57, Lemma 3.2]), and coercive (see the proof of Proposition 3.3
below). To overcome the drawback of the fact that In62C1(Xn;R), we will work with the following
reformulation of problem (3.4) as a variational inequality
8
><
>:hn(v)hn(un)hg0
n(un);vuni0;8v2Xn
un2Xn:(3.35)
This type of denition is commonly used when the Euler-Lagrange functional associated to the equation,
fails to be smooth, but it is the sum between a convex, proper and lower semicontinuous function and
a function of class C1. This method is underlined by Szulkin [100]. According to the terminology from
[100], we refer to a solution of (3.35) as being a critical point ofIn. We will also call unavariational
solution of problem (3.4).
3.2.3 Main result
We are now ready to give the main result of this section.
Theorem 3.3. Problem (3.4), with 'ngiven by relation (3.32), has a unique variational solution for
each integer n>1, provided that pn2[2;1), which is nonnegative in
, sayun. Moreover, under the
supplementary assumption that limn!1pn=1, the sequencefungXnconverges uniformly in
to
= dist(
;@
).

3. Torsional creep problems 62
3.2.4 A -convergence result
For each integer n>1, consider the functional Jn:L1(
)![0;1], dened by
Jn(u) =8
><
>:Z

1
'n(1)n(jruj)dx; ifu2Xn;
+1; otherwise:
The main result of this section gives the following -convergence result for the sequence fJng.
Theorem 3.4. DeneJ1:L1(
)![0;1]by
J1(u) =8
><
>:0; ifu2W1;1(
)\
\q>1W1;q
0(
)
andjru(x)j1;a.e.x2
;
+1;otherwise:
Then (L1(
))lim
n!1Jn=J1.
Proof. Letu2L1(
) andfungL1(
) be such that un!uinL1(
). We may assume without
loss of generality that for each integer n>1, we have un2Xnand
lim inf
n!1Jn(un) = lim
n!1Jn(un)<1: (3.36)
SinceXnW1;pn
0(
), we have un2W1;pn
0(
), for each integer n>1.
Letx2
be a Lebesgue point for ru2(L1(
))N. For any ball Br(x)
and for any integer
n>1, such that pn2, we have, by H older's inequality,
Z
Br(x)jrun(y)jdykjrunjkLpn(
)kBr(x)k
Lpn
pn1(
): (3.37)
We also have
kBr(x)k
Lpn
pn1(
)=jBr(x)jpn1
pn: (3.38)
On the other hand, note that
Z

jrunjpndxZ

(ejrunjpn1)dx=pneJn(un);
or
kjrunjkLpn(
)(pne)1=pn[Jn(un)]1=pn: (3.39)
By (3.37), (3.38) and (3.39), we obtain
Z
Br(x)jrun(y)jdyjBr(x)jpn1
pn(pne)1=pn[Jn(un)]1=pn;
which in view of (3.36), implies
lim sup
n!1Z
Br(x)jrun(y)jdyjBr(x)j: (3.40)

3. Torsional creep problems 63
Letq1 be an arbitrary real number. For each integer n > 1, such that q < pn, using H older's
inequality, we have
Z

jrun(x)jqdxZ

jrun(x)jpndxq
pnj
jpnq
pn
[pneJn(un)]q
pnj
jpnq
pn;
and thus,
kjrunjkLq(
)[pneJn(un)]1=pnj
j1
q1
pn:
We obtain that the sequence frungis bounded in Lq(
;RN), for anyq1. It follows that fung
is bounded in W1;q
0(
), and thus we may extract a subsequence, still denoted by un, such that un
converges weakly to uinW1;q
0(
). A well-known weak lower semicontinuity result implies
Z
Br(x)jru(y)jdylim inf
n!1Z
Br(x)jrun(y)jdy
lim sup
n!1Z
Br(x)jrun(y)jdy;
which, in view of (3.40), yields
1
jBr(x)jZ
Br(x)jru(y)jdy1;8r>0:
Since almost every x2
is a Lebesgue point for ru, passing to the limit r!0+in the above inequality,
yieldsjru(x)j1 for a.e.x2
.
Now, since u2\q>1W1;q
0(
), we deduce that
Z

jrujqdx1(q)Z

jujqdx;8q>1;
where by1(q) we have denoted the rst eigenvalue of the q-Laplacian. In view of [68, Lemma 1.5],
lettingq!1 in the above relation, we get
1kjrujkL1(
)kk1
L1(
)kukL1(
);
and, thus, we deduce u2W1;1(
) and, consequently, u2W1;1(
)\
\q>1W1;q
0(
)
. It follows that
J1(u) = 0 and, thus, we obtain
J1(u)lim inf
n!1Jn(un):
It remains to prove the existence of a recovery sequence for the -limit. Let u2L1(
). Note,
ifJ1(u) = +1there is nothing to prove, because the inequality holds for any sequence un!u
strongly in L1(
). On the other hand, if J1(u)<1, we must have J1(u) = 0 and, consequently,

3. Torsional creep problems 64
u2W1;1(
)\
\q>1W1;q
0(
)
andjru(x)j1 for a.e.x2
. For each integer n>1, deneun:=u
and note that, we have
lim sup
n!1Jn(un) = lim sup
n!1Z

1
'n(1)n(jrun(x)j)dx
lim sup
n!1Z

1
'n(1)n(1)dx
= lim sup
n!1e1
pnej
j= 0 =J1(u):
The proof of Theorem 3.4 is complete. 
3.2.5 Proof of the main result
We start by establishing the following result.
Proposition 3.3. For each integer n>1, such that pn2, problem (3.35) has a unique nonnegative
variational solution.
Proof. Let integer n>1 withpn2 be arbitrary, but xed.
Existence. We show that there exists un2Xnsuch that
1<n:= inffIn() :2Xng=In(un): (3.41)
In order to do that we show rst that Inis coercive, i.e. lim kuk1;n!1;u2XnIn(u) =1. The fact
thatkuk1;n!1 assures that at least one of the following situations occurs: kjrujkn!1 or
kukn!1 . In view of Lemma 3.6 and [29, Lemma C.9], we deduce that for each 2Xnwith
kjrjkn>1, we have
In()1
epnkjrjkpn
nCnkjrjkn; (3.42)
whereCnstands for a positive constant. We deduce that Inis coercive in the case when u2Xnand
kjrujkn!1 . Assume further that u2Xnwithkukn!1 . Then for some q > N , by Lemma
3.6 and Morrey's inequality, we have XnW1;q
0(
)L1(
). Next, note that by [29, Lemma C.9],
we have Z

n(juj)dxkukpn
n;8u2Xnwithkukn>1:
Thus, we get
j
jn(kukL1(
))kukpn
n;8u2Xnwithkukn>1;
which implies that
lim
kukn!1;u2Xnn(kukL1(
)) =1;
or, taking into account the properties of N-functions, this yields to the conclusion that kukL1(
)!1 ,
provided that u2Xnwithkukn!1 . Invoking again the fact that XnW1;q
0(
)L1(
) and

3. Torsional creep problems 65
Morrey's inequality, we nd that kjrujkn!1 and, consequently, relation (3.42) continues to hold
true. From the above proof we get the coercivity of Inin this second case, too.
On the other hand, we remark that, if 2Xn, withkjrjkn1, we have
In()CnkjrjknCn:
Overall, it is clear that Inis bounded from below on Xnand, consequently, n>1.
LetfukgXnbe a minimizing sequence, In(uk)!nask!1:The coercivity of Inimplies
thatfukgis bounded in XnW1;n(
) which is the dual of a separable Banach space. Hence, by
Proposition 3.2, we can extract a subsequence (not relabeled) of fukgwhich converges in the weak?
topology of W1;n(
) to, say, un2Xn. We seek to show that unis as claimed in (3.41). By the weak?
lower semicontinuity of hnand the fact that the embedding of W1;n(
) ontoL1(
) is compact, we
getIn(un)lim inf
k!1In(uk) =n. Thus,In(un) =n.
It remains to show that unis a critical point of functional Inand, thus, a variational solution of
problem (3.35). To this end, recall that In(v)In(un) or, equivalently, hn(v)gn(v)hn(un)gn(un),
for allv2Xn. Thus,
hn(v)hn(un)gn(v)gn(un) =hg0
n(un);vuni;8v2Xn:
Uniqueness. Assume there are two variational solutions of problem (3.35), say u1,u22Xn. It
follows that
Z

n(jrvj)dxZ

n(jru1j)dx'n(1)Z

(vu1)dx0;8v2Xn; (3.43)
and Z

n(jrvj)dxZ

n(jru2j)dx'n(1)Z

(vu2)dx0;8v2Xn; (3.44)
where n(t) =ejtjpn1 for eacht0. Takingv=u1+u2
2in (3.43) and (3.44) we get
Z

nru1+ru2
2
dxZ

n(jru1j)dx'n(1)Z

u2u1
2dx0;
and Z

nru1+ru2
2
dxZ

n(jru2j)dx'n(1)Z

u1u2
2dx0:
Adding the last two inequalities and taking into account that function hnis convex, we obtain
Z

nru1+ru2
2
dx=1
2Z

n(jru1j)dx+Z

n(jru2j)dx
: (3.45)
Next, let us use the fact that function R3t7!etis convex and of class C1overRin order to deduce
that
etea+ea(ta);8t; a2R:
Thus, we nd
ejtjpnejajpn+ejajpn(jtjpnjajpn);8t; a2R:

3. Torsional creep problems 66
In particular the above inequality yields
ejru1jpn1eru1+ru2
2pn
1 +eru1+ru2
2pn
jru1jpnru1+ru2
2pn
;in
;
and
ejru2jpn1eru1+ru2
2pn
1 +eru1+ru2
2pn
jru2jpnru1+ru2
2pn
;in
:
Integrating the last two inequalities over
, then adding the resulting relations and taking into account
(3.45) we deduce
0Z

eru1+ru2
2pn
jru1jpn+jru2jpn2ru1+ru2
2pn
dx:
By Clarkson's inequality with p2 (see, e.g. [1, Lemma 2.37]), we know that
jjp+jjp2+
2p
+ 2
2p
;8; 2RN;
and combining that with the above estimate, we get
0Z

eru1+ru2
2pnjru1ru2jpn
2pn1dx1
2pn1Z

jru1ru2jpndx:
Consequently,kjru1ru2jkLpn(
)= 0, which in view of the fact that XnW1;pn
0(
)Lpn(
),
yieldsu1=u2.
Finally, note that since In(un)In(junj) andInpossesses a unique critical point, we must have
un=junj0. The proof of Proposition 3.3 is now complete. 
In order to go further, we show that the result proved by Payne & Philippin in [92] for problem
(3.1) continues to hold true for problem (3.4) with 'ngiven by relation (3.32). More exactly, we show
the following proposition.
Proposition 3.4. For each integer n > 1such thatpn2, letun2Xnbe the (unique) variational
solution of problem (3.35). If limn!1pn=1, then limn!1R

undx=R

dx.
The proof of this proposition will be based on two auxiliary results which are established in the
next two lemmas.
Lemma 3.7. The sequenceZ

undx
is bounded.
Proof. For everypn2, H older's inequality gives
Z

undxZ

upnndx1=pn
j
j(pn1)=pn:

3. Torsional creep problems 67
Denote by 1(pn), the rst eigenvalue of the pn-Laplacian (see Lindqvist [79] for more details), given
by
1(pn) := inf
v2W1;pn
0(
)nf0gZ

jrvjpndx
Z

jvjpndx:
It is easy to check that actually
1(pn) := inf
v2W1;pn
0(
)nf0gZ

n(jrvj)dx
Z

jvjpndx;
where n(t) =ejtjpn1, for allt2R. The above estimates and the fact that XnW1;pn
0(
) imply
Z

undxpn
j
jpn1Z

n(jrunj)dx
1(pn):
On the other hand, relation (3.35) with v= 0, gives
Z

n(jrunj)dx'n(1)Z

undx;
and thus, combining the last two relations, we get
Z

undxpn
j
jpn1
1(pn)'n(1)Z

undx;
orZ

undxj
j['n(1)]1=(pn1)
1(pn)1=(pn1)=j
je1=(pn1)p1=(pn1)
n
(pnp
1(pn))pn=(pn1):
On the other hand, by [68, Lemma 1.5], we know that lim n!1[1(pn)]1=pn=kk1
L1(
);which implies
that the right-hand side in the last inequality above is bounded and, consequently,Z

undx
is
bounded. The proof of Lemma 3.7 is complete. 
Lemma 3.8. There exists u12W1;1(
)\
\q>1W1;q
0(
)
withkru1kL1(
;RN)1and a subse-
quence offung(not relabeled), such that un!u1uniformly in
:
Proof. Letq>1 be an arbitrary real number. Consider that integer n>1 is suciently large such
thatpn>q. Using H older's inequality, and recalling the fact that (3.35) holds true, we deduce that
Z

jrunjqdxZ

jrunjpndxq
pnj
jpnq
pn
Z

n(jrunj)dxq
pnj
jpnq
pn

'n(1)Z

undxq
pnj
jpnq
pn:

3. Torsional creep problems 68
By Lemma 3.7, we deduce that there exists a positive constant Csuch thatZ

undxCfor all integers
n > 1 for which pn2. Thus, for all integers n > 1 suciently large, such that pn> q, we must
havekrunkLq(
;RN)C1=pnj
j(pnq)=(qpn)e1=pnp1=pnn:We nd that the sequence frungis bounded in
Lq(
;RN), for anyq>1. Since by Lemma 3.7 the sequence fungis also bounded in L1(
), we obtain,
in view of the Poincar e-Wirtinger inequality, that fungis bounded in Lq(
). Hence, there exists a
subsequence (not relabeled), and u12W1;q
0(
);such thatun*u1weakly inW1;q
0(
) andun!u1
strongly in Lq(
):However, since q2 was arbitrary, the compactness of the embedding of W1;q
0(
)
intoC0;(
) (0<< 1) forq>N (one can choose, e.g., = 1N=q), allows us to conclude that in
factunconverges to u1uniformly in
.
Finally, in view of Proposition 1.2 (with X=L1(
),Fn=Jn,F1=J1,zn=un) and Theorem
3.4, we conclude that u1is a minimizer for J1which, in particular, means that u12W1;1(
)\
\q>1W1;q
0(
)
andkru1kL1(
;RN)1:This concludes the proof of Lemma 3.8. 
Proof of Proposition 3.4. Fix an arbitrary subsequence of fung, still denoted by fung. Similar
arguments as those used in the proof of Lemma 3.8 can be considered to prove that this subsequence
contains, in its turn, a subsequence, say funkg, which converges uniformly in
to a certain limit u12
W1;1(
)\
\q>1W1;q
0(
)
withkru1kL1(
;RN)1. In order to get the conclusion of Proposition 3.4
it is enough to establish that lim k!1R

unkdx=R

dx. In other words, we will show that the limit
of all possible subsequences of fR

undxgisR

 dx and, consequently, the limit of the full sequence
should also beR

dx.
In the sequel, for simplicity, we will write uninstead ofunk.
Since2Xn;jr(x)j= 1 for a.e. x2
, andunis a minimizer of IninXn, we deduce that for
each integer n>1 for which pn2, we have
In(un)In() =Z

n(1)
'n(1)dxZ

dx;
or,Z

dxZ

undx+j
j(e1)
pne;8n>1 s:t:pn2:
Consequently, we nd Z

dxlim
n!1Z

undx=Z

u1dx: (3.46)
Next, we observe that for each x2
andy2@
, such thatjxyj=(x), we have
u1(x) =u1(x)u1(y)jxyjsup
z2[x;y]jru1(z)j(x): (3.47)
Integrating over
, we get
lim
n!1Z

undx=Z

u1dxZ

dx:

3. Torsional creep problems 69
Recalling (3.46), we deduce that lim
n!1Z

undx=Z

 dx, which concludes the proof of Proposition
3.4.
We are now ready to give the proof of Theorem 3.3.
Proof of Theorem 3.3. As in Proposition 3.4 we x an arbitrary subsequence of the solutions
fung(not relabeled). Similar arguments as those used in Lemma 3.8 ensure that fungconverges
uniformly to a certain limit u12W1;1(
)\
\q>1W1;q
0(
)
withkru1kL1(
;RN)1. Hence, it
just remains to see that u1=. Notice that, since fungis arbitrary, this means that is indeed the
limit of the full sequence fung. Recall that, by (3.47), we have u1(x)(x), for eachx2
. Further,
since we have un(x)0 for a.e.x2
and for every integer n>1 for which pn2, we deduce that
u1(x)0 for a.e.x2
. Finally, applying Proposition 3.4 and taking into account the fact that
un!u1uniformly in
, we nd that
Z

dx = lim
n!1Z

undx=Z

u1dx:
Recalling the continuity of andu1, the last equalities yield u1=. The proof of Theorem 3.3 is
complete. 
3.3 On a family of torsional creep problems in Finsler metrics
In this section, for each real number p2(N;1), we consider the following problem
8
><
>:div((x;u)H(ru)p2H(ru)) =f; x2
;
u= 0; x 2@
;(3.48)
whereH:RN![0;1) is a Finsler norm (see Section 3.3.2 for details), :
R!(0;1) is a
continuous function for which there exist two positive constants ;, such that
0<(x;t)<+1;8×2
;8t2R; (3.49)
f:
!(0;1) is a given continuous function and H:RN!RNis dened by
Hi() :=@
@i1
2H()2
;82RN;8i2f1;:::;Ng: (3.50)
In the particular case when (x;t) andf(x) are positive constant functions and H(
) =j
jis the
Euclidian norm on RN, equation (3.48) reduces to problem (3.1). Consequently, we can regard the
general case of problem (3.48) as an extension of the classical model to the situation when we deal
with an anisotropic material or with the case when the Euclidian distance in
is distorted due to the
presence of the Finsler norm (see Belloni & Kawohl [11] or Belloni, Kawohl, & Juutinen [12] for similar
interpretations of the use of Finsler norms). Moreover, the motivation of the presence of a nonconstant
function(x;u), depending on u, in the divergence operator involved in problem (3.48) is to take into

3. Torsional creep problems 70
account the reaction of this equation to its own state (see, e.g. Chipot [28, p. 160]). In particular, the
temperature which was assumed to be constant in Kachanov's model may vary in this new situation.
Our goals in this general setting will be, rst, to show the existence of a solution of equation (3.48),
for eachp > N , and next, to prove the uniform convergence of the family of solutions, as p!1 , to
a distance function to the boundary of
, which takes into account the Finsler norm involved in the
equation, i.e. H(x) := infy2@
H0(xy), for eachx2
(noteH0stands for the dual norm of H; the
denition of H0is provided in Section 3.3.2).
3.3.1 Main result
The notion of a solution for equation (3.48) will be understood in the weak sense. More precisely, we
work under the following denition.
Denition 3.4. We say that upis a weak solution of problem (3.48), if up2W1;p
0(
)and it satises
the following relation
Z

(x;up)H(rup)p2hH(rup);r'idx=Z

f'dx;8'2W1;p
0(
): (3.51)
The main results of this section are given by the following theorems.
Theorem 3.5. Assume that condition (3.49) is fullled. Then, for each p2(N;1), problem (3.48)
has a weak solution up2W1;p
0(
), such that up(x)0, for a.e.x2
.
Theorem 3.6. Assume that condition (3.49) is fullled. Let fpngn(N;1)be a sequence of real
numbers satisfying lim
n!1pn=1. For each n > 1, denote by upn2W1;pn
0(
) a weak, nonegative
solution of problem (3.48) with p=pn. Then, the sequence fupngnconverges uniformly in
to the
distance function to the boundary of domain
, given byH(x) := infy2@
H0(xy), for eachx2
.
3.3.2 Finsler norms: denition, properties, examples
LetH:RN![0;1) be a convex function of class C2(RNnf0g), even and homogeneous of degree 1,
i.e.
H(t) =jtjH();8t2R; 2RN:
We will refer to Has being a Finsler norm .
Set
K:=fx2RN:H(x)1g;
and
H(x) := sup
2Khx;i:

3. Torsional creep problems 71
We will refer to Has being the support function of K. It is easy to check that H:RN![0;1) is a
convex homogeneous function, and actually, a Finsler norm, too. We will call HandHpolar to each
other, in the sense
H(x) := sup
6=0hx;i
H();
and
H(x) := sup
6=0hx;i
H():
The above relations yield
jhx;ijH(x)H();8x; 2RN: (3.52)
Examples of Finsler norms.
1) The Euclidian norm: H(x) =jxj=NP
i=1jxij21=2
;
2)H(x) =hAx;xi, whereAis a symmetric, positive denite NNmatrix;
3) Thep-normH(x) =NP
i=1jxijp1=p
, withp2(1;1);
4)H(x) =rq
x4
1+:::+x4
N+x2
1+:::+x2
N.
We recall some important properties regarding functions HandHthat will be useful in our
subsequent analysis
H(rH()) = 1 and H(rxH(x)) = 1; (3.53)
hrxH(x);xi=H(x) andhrH();i=H();8xand2RN: (3.54)
We refer to [9] for the proofs of the above relations and to [30, p. 352], for some similar relations
obtained in the case when His a more general Finsler norm.
Further, let us also recall the so-called fundamental inequality regarding Finsler norms, namely for
eachx2RN, we have
h;rH(x)iH();86= 0; (3.55)
and equality holds if and only if x=, for some0 (see [3, Theorem 1.2.2, relation (1.2.3)] for
more details).
Since any two norms are equivalent on RN, we infer that for Hdened as above, there exist two
positive constants aandb, such that
ajxjH(x)bjxj;8x2RN; (3.56)
(see, e.g. [10] or [4]).

3. Torsional creep problems 72
Using a Finsler norm, we can dene, for each real number p2(1;1) a dierential operator which
generalizes the classical p-Laplacian, namely
Qpu:=NX
i=1@
@xi[H(ru)p2Hi(ru)];
whereHwas dened in relation (3.50). Note that Qpis a particular case of the dierential operator
involved in equation (3.48) which is obtained in the particular case when (x;u) = 1. Moreover, it is
useful to observe that H() =H()rH().
It is known (see e.g. Belloni, Ferone & Kawohl [10] or Belloni, Kawohl & Juutinen [12]) that for each
real number p2(1;1), the minimum of the Rayleigh quotient associated to the eigenvalue problem
8
><
>:Qpv=jvjp2v ifx2
v= 0 if x2@
;(3.57)
i.e.
1(p) := inf
v2W1;p
0(
)nf0gZ

H(rv)pdx
Z

jvjpdx>0;
stands for the lowest eigenvalue of problem (3.57) whose corresponding eigenfunctions are minimizers
of1(p) that do not change sign in
. Moreover, for p>N a minimizer is C1-H older continuous.
In particular, for each p>1, we have
Z

H(rv)pdx1(p)Z

jvjpdx;8v2W1;p
0(
): (3.58)
Further, dene the distance function to the boundary of
with respect to the dual of the Finsler norm
H, i.e.H:
![0;1) given by
H(x) := inf
y2@
H0(xy);8×2
:
Recall that His Lipschitz continuous and satises H(rH(x)) = 1 for a.e. x2
(see, e.g. [12,
Section 3] or [83] for a more involved discussion regarding the distance function to the boundary in
Finsler metrics). Dene also
1:=kH(rH)kL1(
)
kHkL1(
)=kHk1
L1(
):
By [12, Lemma 3.1], we know that
lim
p!1(1(p))1=p= 1: (3.59)
Note that, in the particular case when we work with the Euclidian norm, this result was obtained by
Juutinen, Lindqvist & Manfredi [68] and Fukagai, Ito & Narukawa [54].

3. Torsional creep problems 73
3.3.3 Proof of Theorem 3.5
We start by establishing some auxiliary results which will be useful in obtaining the conclusion of
Theorem 3.5.
Lemma 3.9. For eachv2Lp(
), problem
8
><
>:div((x;v)H(ru)p2H(ru)) =f; x2
;
u= 0; x 2@
(3.60)
has a unique weak solution u2W1;p
0(
), i.e.
Z

(x;v)H(ru)p2hH(ru);r'idx=Z

f'dx;8'2W1;p
0(
); (3.61)
which satises u0a.e. in
.
Proof. Step 1: Existence . Fixv2Lp(
). By hypotheses (3.49), we get (x;v)2L1(
).
Consider the energy functional associated to problem (3.60), J:W1;p
0(
)!R, dened by
J(u) =1
pZ

(x;v)H(ru)pdxZ

fudx:
Standard arguments imply that J2C1(W1;p
0(
);R) with the derivative given by
hJ0(u);'i=Z

(x;v)H(ru)p2hH(ru);r'idxZ

f'dx;8u;'2W1;p
0(
):
Thus, the weak solutions of problem (3.60) are exactly the critical points of J.
By relations (3.49), (3.56) and (3.58) and using H older's inequality, we deduce that for each p2
[2;+1) andu2W1;p
0(
), we have
J(u)ap
pZ

jrujpdxkfkLp0(
)kukLp(
)
ap
pkukp
W1;p
0(
) kfkLp0(
)(1(p))1=pbkukW1;p
0(
);
wherep0=p
p1is the conjugate exponent of p. The above estimates show that Jis coercive. On the
other hand, it is standard to check that Jis weakly lower semi-continuous. Then, the Direct Method
in the Calculus of Variations (see Theorem 1.1) guarantees the existence of a global minimum point of
J, sayu2W1;p
0(
). It is also standard to prove that uis a weak solution of problem (3.60).
Step 2: Uniqueness . Assume there are two weak solutions of problem (3.60), say u1; u22W1;p
0(
),
which means that u1andu2are critical points for functional J. The standard regularity theory of
elliptic operators assures that u1; u22C1;(
), for some 2(0;1). Dene

+:=fx2
:u1(x)>u 2(x)g:

3. Torsional creep problems 74
The continuity of u1andu2assures that
+is an open subset of
. Since hJ0(u1);'i= 0 and
hJ0(u2);'i= 0, for all '2W1;p
0(
), working with the extension of ( u1u2)+to
by zero outside
+
as a test function, it follows that
Z

+(x;v)H(ru1)p2hH(ru1);r(u1u2)idxZ

+f(u1u2)dx= 0;
and Z

+(x;v)H(ru2)p2hH(ru2);r(u1u2)idxZ

+f(u1u2)dx= 0:
Subtracting these two equalities term by term, we obtain
Z

+(x;v)hH(ru1)p2H(ru1)H(ru2)p2H(ru2);ru1ru2idx= 0:
By the strict convexity of the mapping RN3!Hp(), we haveru1(x) =ru2(x) for a.e.x2
+.
Sinceu1=u2on@
+, we nd that
+has measure zero. Similarly, the set
:=fx2
:u1(x)<
u2(x)ghas measure zero which yields u1=u2.
Step 3: Nonnegativity . Finally, note that since J(u)J(juj), for allu2W1;p
0(
) andJpossesses
a unique critical point, we must have u=juj0 a.e. in
. The proof of Lemma 3.9 is complete. 
Next, for each v2C(
)Lp(
), letu=T(v)2W1;p
0(
) be the unique weak solution of problem
(3.60) given by Lemma 3.9. Thus, we can actually introduce an application
T:C(
)!W1;p
0(
);
associating to each v2C(
) the unique weak solution of problem (3.60), denoted by T(v)2W1;p
0(
).
Lemma 3.10. There exists a universal constant Cp>0, which does not depend on v, such that
Z

H(rT(v))pdxCp;8v2C(
): (3.62)
Proof. SinceT(v) is a weak solution of problem (3.60), taking '=T(v) in (3.61), we nd
Z

(x;v)H(rT(v))p2hH(rT(v));rT(v)idx=Z

fT(v)dx:
Using relations (3.49) and (3.54), H older's inequality and (3.58), we deduce
Z

H(rT(v))pdx kfkL1(
)Z

T(v)dx
 kfkL1(
)j
j(p1)=pkT(v)kLp(
)
 kfkL1(
)j
j(p1)=p0
BB@Z

H(rT(v))pdx
1(p)1
CCA1=p
;

3. Torsional creep problems 75
wherej
jstands for the Lebesgue measure of domain
. Thus, we have
Z

H(rT(v))pdx(p1)=p
kfkL1(
)
j
j(p1)=p 1
1(p)1=p:
Taking
Cp:=kfkL1(
)
p=(p1)
j
j1
1(p)1=(p1);
we obtain inequality (3.62). The proof of Lemma 3.10 is complete. 
Remark 3.1. Using Lemma 3.10 and inequality (3.58), it follows that there exists a positive constant
Dp, such that
Z

jT(v)jpdxkfkL1(
)
p=(p1)
j
j1
1(p)p=(p1):=Dp;8v2C(
):
Next, using Morrey's inequality (see [27]) and relation (3.56), we deduce that there exist two positive
constantsEpandEp, such that
kT(v)kL1(
)EpkrT(v)kLp(
)Ep
aC1=p
p:=Ep;8v2C(
):
Lemma 3.11. The mapT:C(
)!W1;p
0(
)is continuous.
Proof. LetfvngC(
) andv2C(
), such thatfvngconverges uniformly to vin
. Setun:=
T(vn), for any positive integer n.
By Lemma 3.10 and relation (3.56), we infer
Z

jrunjpdx=Z

jrT(vn)jp1
apZ

H(rT(vn))pCp
ap;8n;
that is the sequence fungis bounded in W1;p
0(
). It follows that there exists u2W1;p
0(
) such that, up
to a subsequence (still denoted by fung),fungconverges weakly to uinW1;p
0(
) andfungconverges
uniformly to uin
sincep>N . On the other hand, we have unis a weak solution of problem (3.60)
and, thus, by (3.61), we get
Z

(x;vn)H(run)p2hH(run);r'idx=Z

f'dx;8'2W1;p
0(
);8n: (3.63)
Sincefvngconverges uniformly to vin
andis continuous on
R, we nd that (x;vn(x))
converges uniformly to (x;v(x)) on
. For'=unuin (3.63) and taking into account the above
pieces of information, we also nd
lim
n!1Z

(x;vn)H(run)p2hH(run);runruidx= 0;

3. Torsional creep problems 76
and, consequently,
lim
n!1Z

[(x;vn)(x;v)]H(run)p2hH(run);runruidx
+Z

(x;v)H(run)p2hH(run);runruidx
= 0:(3.64)
Taking into account again that (x;vn(x)) converges uniformly to (x;v(x)) in
, by (3.54), (3.62) and
H older's inequality, we deduce
Z

[(x;vn)(x;v)]H(run)p2hH(run);runidx
Z

[(x;vn)(x;v)]H(run)pdx
 k(x;vn)(x;v)kL1(
)Cp!0 asn!1:(3.65)
Similarly, using (3.55) and H older's inequality, we obtain
Z

[(x;vn)(x;v)]H(run)p2hH(run);ruidx
 k(x;vn)(x;v)kL1(
)Z

H(run)p1jhrH(run);ruijdx
 k(x;vn)(x;v)kL1(
)Z

H(run)p1H(ru)dx
 k(x;vn)(x;v)kL1(
)kH(run)p1kLp=(p1)(
)kH(ru)kLp(
)
=k(x;vn)(x;v)kL1(
)kH(run)kp1
Lp(
)kH(ru)kLp(
)
 k(x;vn)(x;v)kL1(
)C(p1)=p
pkH(ru)kLp(
)!0;
asn!1 . Then, the above estimates and (3.65) yield
lim
n!1Z

[(x;vn)(x;v)]H(run)p2hH(run);runruidx= 0: (3.66)
By (3.64) and (3.66), we nd
lim
n!1Z

(x;v)H(run)p2hH(run);runruidx= 0: (3.67)
Using H older's inequality, we have that the application T:W1;p
0(
)!R;
T(w) =Z

(x;v)H(ru)p2hH(ru);rwidx;8w2W1;p
0(
);
is linear and continuous and since fungconverges weakly to uinW1;p
0(
), it follows that
lim
n!1Z

(x;v)H(ru)p2hH(ru);r(unu)idx= 0: (3.68)

3. Torsional creep problems 77
Combining (3.67) and (3.68), we nd
lim
n!1Z

(x;v)hH(run)p2H(run)H(ru)p2H(ru);runruidx= 0:
In particular, (x;v)hH(run)p2H(run)H(ru)p2H(ru);runrui0 and it converges to 0 in
L1(
). Thus, up to a subsequence, (x;v)hH(run)p2H(run)H(ru)p2H(ru);runrui! 0
a.e. in
. Therefore run!rua.e. in
.
On the other hand, the convexity of the mapping RN3!Hp(), yields
Z

(x;v)H(ru)pdxZ

(x;v)H(run)pdx+
pZ

(x;v)H(run)p2hH(run);runruidx;8n:
This estimate and relation (3.67), imply
Z

(x;v)H(ru)pdxlim sup
n!1Z

(x;v)H(run)pdx:
Further, the fact that unconverges weakly to uinW1;p
0(
) and some well-known convexity arguments
yield
lim inf
n!1Z

(x;v)H(run)pdxZ

(x;v)H(ru)pdx:
The last two relations show that
lim
n!1Z

(x;v)H(run)pdx=Z

(x;v)H(ru)pdx: (3.69)
Consider now the sequence fgngninL1(
) dened pointwise in
by
gn(x) :=(x;v(x))H(run(x))p+(x;v(x))H(ru(x))p
2
(x;v(x))Hrun(x)ru(x)
2p
0:
It is clear that gn!(x;v)H(ru)pa.e. in
. Then, by Fatou's Lemma and (3.69), we have
Z

(x;v)H(ru)pdxlim inf
n!1Z

gndx
=Z

(x;v)H(ru)pdx
lim sup
n!1Z

(x;v)Hrunru
2p
dx:
We conclude that, up to a subsequence, fungconverges strongly to uinW1;p
0(
); that means the
application Tis continuous. The proof of Lemma 3.11 is complete. 

3. Torsional creep problems 78
Remark 3.2. SinceW1;p
0(
)is compactly embedded in C(
), forp > N , i.e. the inclusion operator
i:W1;p
0(
)!C(
)is compact, it follows by Lemma 3.11, that the operator S:C(
)!C(
), dened
byS=iTis compact.
Proof of Theorem 3.5. For eachp > N , letEpbe the positive constant given by Remark 3.1. We
have
kS(v)kL1(
)Ep;8v2C(
):
Dene the set in C(
),
BEp(0) :=
v2C(
) :kvkL1(
)Ep
:
Clearly,BEp(0) is a convex, closed subset of C(
) andS(BEp(0))BEp(0):By Remark 3.2, it follows
thatS(BEp(0)) is relatively compact in C(
).
Finally, by Lemma 3.11 and Remark 3.2, we deduce that S:BEp(0)!BEp(0) is a continuous map.
Hence, we can apply Schauder's xed point theorem to obtain that Spossesses a xed point up. This
gives us a weak solution up2W1;p
0(
) of problem (3.48) which is nonnegative in
. The proof of
Theorem 3.5 is nally complete. 
Remark 3.3. For eachp > N , there exists up2W1;p
0(
)C(
)such thatT(up) =up, whereupis
obtained by applying Schauder's xed point theorem, and upis the unique minimizer of the functional
J:W1;p
0(
)!R;
J(u) =1
pZ

(x;up)H(ru)pdxZ

fudx:
3.3.4 Proof of Theorem 3.6
Letfpngn1(N;1) be a sequence of real numbers, such that lim
n!1pn= +1:
For eachn1, we consider upnto be the weak solution of problem (3.48) with p=pn, which is
obtained by applying Schauder's xed point theorem. We have that upnis the unique weak solution of
problem (3.60) with p=pnandv=upn. By Remark 3.3, we have that upnis the unique minimizer of
the functional Jn:W1;pn
0(
)!R;dened by
Jn(u) =1
pnZ

(x;upn)H(ru)pndxZ

fudx:
Lemma 3.12. For eachn1, deneIn;I1:L1(
)![0;1]by
In(u) :=8
><
>:1
pnZ

(x;upn)H(ru)pndx; ifu2W1;pn
0(
);
+1; ifu2L1(
)nW1;pn
0(
):

3. Torsional creep problems 79
and
I1(u) :=8
><
>:0; ifu2X0andH(ru(x))1fora:e:x2
;
+1;otherwise;
whereX0=W1;1(
)\
\q>1W1;q
0(
)
. Then (L1(
))lim
n!1In=I1:
Proof. Letv2L1(
) andfvngL1(
) be, such that vn!vinL1(
). We may assume without
loss of generality that fvngW1;pn
0(
) and
lim inf
n!1In(vn) = lim
n!1In(vn)<1: (3.70)
Letx2
be a Lebesgue point for rv2(L1(
))N. For any ball Br(x)
and for any integer n>1,
such thatpn2, we have, by H older's inequality,
Z
Br(x)H(rvn(y))dykH(rvn)kLpn(
)kBr(x)k
Lpn
pn1(
): (3.71)
Also, we have
kBr(x)k
Lpn
pn1(
)=jBr(x)jpn1
pn: (3.72)
On the other hand, note that Z

H(rvn)pndx1
pnIn(vn)
or
kH(rvn)kLpn(
)1=pnp1=pnn[In(vn)]1=pn: (3.73)
By (3.71), (3.72) and (3.73), we obtain
Z
Br(x)H(rvn(y))dy1=pnpn1=pn[In(vn)]1=pnjBr(x)jpn1
pn;
which in view of (3.70) implies
lim sup
n!1Z
Br(x)H(rvn(y))dyjBr(x)j: (3.74)
Letq1 be an arbitrary real number. For each n>1, such that q<pn, using H older's inequality, we
have
Z

H(rvn)qdxZ

H(rvn)pndxq
pnj
jpnq
pn
q=pnpq=pnn[In(vn)]q=pnj
jpnq
pn
and thus, using (3.56)
akjrvnjkLq(
)kH(rvn)kLq(
)[1pnIn(vn)]1
pnj
j1
q1
pn:

3. Torsional creep problems 80
We obtain that the sequence frvngis bounded in Lq(
;RN), for anyq1. It follows that the sequence
fvngis bounded in W1;q
0(
), and thus we may extract a subsequence, still denoted by fvng, such that
vnconverges weakly to vinW1;q
0(
). In particular, we nd that v2\q>1W1;q
0(
). On the other hand,
a well-known weak lower semicontinuity result implies
Z
Br(x)H(rv(y))dylim inf
n!1Z
Br(x)H(rvn(y))dylim sup
n!1Z
Br(x)H(rvn(y))dy
which, in view of (3.74), yields
1
jBr(x)jZ
Br(x)H(rv(y))dy1;8r>0:
Since almost every x2
is a Lebesgue point for rv, passing to the limit as r!0+in the above
inequality, yields H(rv(x))1, for a.e.x2
. Now, since v2\q>1W1;q
0(
), we deduce, by (3.58),
that Z

H(rv)qdx1(q)Z

jvjqdx;8q>1:
In view of (3.59) the above relation implies that
1kH(rv)kL1(
)1kvkL1(
);
and, thus, we deduce v2W1;1(
) and, consequently, v2X0. It follows that I1(v) = 0 and, thus, we
obtain
I1(v)lim inf
n!1In(vn):
It remains to prove the existence of a recovery sequence for the -limit. Let v2L1(
). Note, if
I1(v) = +1there is nothing to prove, because the inequality holds true for any sequence vn!v
strongly in L1(
). On the other hand, if I1(v)<1, we must have I1(v) = 0 and, consequently,
v2X0\q>1W1;q
0(
) andH(rv(x))1 ,for a.e.x2
. For each integer n>1, denevn:=vand
note that we have
lim sup
n!1In(vn) = lim sup
n!11
pnZ

(x;upn)H(rv(x))pndx
lim sup
n!1
pnj
j= 0 =I1(v):
The proof of Lemma 3.12 is complete. 
Proposition 3.5. For eachn2N, letupn2W1;pn
0(
)be the weak solution of the problem (3.48) with
p=pn, given by Schauder's xed point theorem. Then
lim
n!1Z

fupndx=Z

fHdx:
The proof of this proposition will be based on two auxiliary results, which are established in the
next two lemmas.

3. Torsional creep problems 81
Lemma 3.13. The sequencesZ

fupndx
andZ

upndx
are bounded.
Proof. First, we prove that the sequenceZ

fupndx
is bounded. For every pn2, using H older's
inequality, we have
Z

fupndxkfkL1(
)Z

upndxkfkL1(
)Z

jupnjpndx1=pn
j
jpn1
pn:
Takingp=pnandv=upnin (3.58), we have
Z

jupnjpndxZ

H(rupn)pndx
1(pn):
Combining the above two inequalities, we deduce
Z

fupndxpn
kfkpn
L1(
)Z

H(rupn)pndx
1(pn)j
jpn1: (3.75)
Sinceupnis a weak solution for problem (3.48) with p=pn, relation (3.51) with '=upngives
Z

(x;upn)H(rupn)pn2hH(rupn);rupnidx=Z

fupndx;
and taking into account (3.54) and (3.49), it follows that
Z

H(rupn)pndx=Z

H(rupn)pn2hH(rupn);rupnidx1
Z

fupndx: (3.76)
By inequalities (3.75) and (3.76), we get
Z

fupndxpn1
kfkpn
L1(
)1
1(pn)j
jpn11

orZ

fupndxkfkpn=(pn1)
L1(
)1=(pn1)

pnp
1(pn)pn=(pn1)j
j:
On the other hand, by (3.59) we know that lim
n!1pnp
1(pn) =kHk1
L1(
);which implies that the
right-hand side in the last inequality above is bounded and, consequently,Z

fupndx
is bounded.
Finally, since f:
!(0;1) is continuous we deduce thatZ

upndx
is bounded, too. The proof
of Lemma 3.13 is complete. 
Lemma 3.14. There exists u12X0withu10in
andkH(ru1)kL1(
)1and a subsequence
offupng(not relabeled), such that upn!u1uniformly in
.

3. Torsional creep problems 82
Proof. Fixq > N an arbitrary real number. Since lim
n!1pn= +1, it follows that q < pn, for
suciently large n2N:
For eachq<pn, using H older's inequality, recalling the fact that hJ0
n(upn);upni= 0 and taking into
account (3.76), we deduce
Z

H(rupn)qdxZ

H(rupn)pndxq
pnj
jpnq
pn
1
Z

fupndxq
pnj
jpnq
pn
1
q=pnZ

fupndxq
pnj
jpnq
pn
By Lemma 3.13, there exists a positive constant M, such that
Z

fupndxM;
for alln2Nsuciently large. Thus, using also (3.56), for such n2N, we must have
akjrupnjkLq(
)kH(rupn)kLq(
)1=pnM1=pnj
j1=q1=pn:
Thus,frupngnis uniformly bounded in Lq(
;RN). The fact that q>N guarantees that the embedding
ofW1;q
0(
) intoC(
) is compact. Taking into account the re
exivity of the Sobolev space W1;q
0(
), it
follows that there exists a subsequence (not relabeled) of fupngand a function u12C(
) such that
upn*u1weakly inW1;q
0(
) andupn!u1uniformly in
. Moreover, the fact that upn0 a.e. in

for eachpn>N, implies that u10 a.e. in
.
Finally, since there exists u1such thatu1= lim
n!1upn, inL1(
) in view of Proposition 1.2 (with
X=L1(
),Fn=In,F1=I1,zn=upn) and Lemma 3.12 (and taking into account that for each
positive integer nthe minimizer upnofJnminimizesIn, too), we conclude that u1must be a minimizer
forI1and, in particular kH(ru1)kL1(
)1 andu12X0. This concludes the proof of Lemma 3.14.

Proof of Proposition 3.5. Fix an arbitrary subsequence of fupng, still denoted byfupng. Similar
arguments as those used in the proof of Lemma 3.14 can be considered to prove that this subsequence
contains, in its turn, a subsequence, say fupnkg, which converges uniformly in
to a certain limit
u12X0withkH(ru1)kL1(
1. In order to get the conclusion of Proposition 3.5, it is enough
to establish that lim k!1R

fupnkdx=R

fHdx. In other words, we will show that the limit of all
possible subsequences of fR

fupndxgisR

fHdxand, consequently, the limit of the full sequence
should also beR

fHdx.
In the sequel, for simplicity, we will write upninstead ofupnk.
SinceH2X0\q>1W1;q
0(
) andH(rH(x)) = 1, for a.e. x2
, andupnis a minimizer of Jnin

3. Torsional creep problems 83
W1;pn
0(
), we deduce that for each positive integer n2N, we have
Jn(upn)Jn(H) =1
pnZ

(x;upn)H(rH(x))pndxZ

fHdx
=1
pnZ

(x;upn)dxZ

fHdx

pnj
jZ

fHdx:
Taking into account that pn!1 asn!1 , the above estimates imply
Z

fHdxlim inf
n!1Z

fupndx=Z

fu1dx: (3.77)
Next, for each x2
xy2@
, such that H0(xy) =H(x). Deneh: [0;1]!Rbyh(t) :=
u1(tx+ (1t)y). By the mean value theorem, we deduce that there exists tx2(0;1) such that
h0(tx) =h(1)h(0);
or
u1(x) =u1(x)u1(y) =h(1)h(0) =hru1(txx+ (1tx)y);(xy)i;
and by (3.52), we deduce
u1(x)H0(xy) sup
z2[x;y]H(ru1(z))H(x): (3.78)
Multiplying by fand integrating over
, we get
lim
n!1Z

fupndx=Z

fu1dxZ

fHdx:
Recalling (3.77), it follows that lim
n!1Z

fupndx=Z

fHdx:Thus, the proof of Proposition 3.5 is
complete. 
Proof of Theorem 3.6. As in Proposition 3.5, we x an arbitrary subsequence of the solutions
fupng(not relabeled). Similar arguments as those used in Lemma 3.14 ensure that fupngconverges
uniformly to a certain limit
u12X0withkH(ru1)kL1(
)1:
Hence, it just remains to see that u1=H. Notice that, since fupngis arbitrary, this means that H
is indeed the limit of the full sequence fupng. Recall that by (3.78) we have u1(x)H(x), for each
x2
. Further, since we have upn(x)0 for a.e.x2
and for every integer n>1, for which pn2,
we deduce that u1(x)0, for a.e.x2
. Finally, applying Proposition 3.5 and taking into account
the fact that upn!u1uniformly in
, we nd that
Z

fHdx= lim
n!1Z

fupndx=Z

fu1dx:
Recalling the continuity of f,Handu1, the last equalities yield u1=H.
The proof of Theorem 3.6 is complete. 

Chapter 4
Final comments and further directions
of research
In this chapter, we recall some open problems from the topic of our thesis that we consider to be useful
in our further research. The role of these problems will be both to guide our further research and to
impose high standards of research in the near future.
4.1 \Can one hear the shape of a drum?"
Let
1and
2be two bounded domains in R2; suppose that the eigenvalues of the operator
(with
homogeneous Dirichlet boundary conditions) are the same for
1and
2.Are
1and
2isometric?
This problem has been nicknamed by M. Kac [69]: \Can one hear the shape of a drum?"
If
1is a disk, then the answer is positive.
If
1is a domain with corners, then the answer is negative (see Gordon, Webb, & Wolpert [60]).
The problem of Kac is still open for smooth domains .
The above information regarding this problem can be found in Brezis [21, p. 321]
4.2 \Hot spots conjecture"
Let
RNbe a bounded domain with smooth boundary denoted by @
. Consider the eigenvalue
problem for
with homogeneous Neumann boundary conditions, i.e.
8
><
>:
u=u in
@u
@= 0 on @
;(4.1)
84

4. Further directions of research 85
wherestands for the unit outer normal to @
and@

@denotes the normal derivative. It is known that
the rst positive eigenvalue of problem (4.1) (which actually represents the second eigenvalue of the
problem) has the following variational characterization
1:= inf
v2H1(
)nf0g;Z

vdx = 0Z

jrvj2dx
Z

v2dx:
Moreover, the above \inf" is actually achieved for some u12H1(
)nf0gwithR

u1dx= 0, which
proves to be an eigenfunction corresponding to the eigenvalue 1. The folowing question is open: does
u1attain its extrema only at the boundary of
?
4.3 The spectrum of the p-Laplace operator when p6= 2
Let
RNbe a bounded domain with smooth boundary denoted by @
and letp2(1;1) be a real
number. Consider the eigenvalue problem for
pwith homogeneous Dirichlet boundary conditions,
i.e. 8
><
>:
pu=jujp2uin
u= 0 on @
:(4.2)
ForN2andp6= 2the complete description of the set of all eigenvalues for problem
(4.2) is an open problem . It is known (see, e.g. [56]) that the Ljusternik-Schnirelman theory ensures
the existence of a nondecreasing and unbounded sequence of positive eigenvalues, but in general this
theory does not provide all eigenvalues. There are many other open questions concerning the set of
eigenvalues of the nonlinear p-Laplacian. We mention a few of them:
Are there other eigenvalues dierent from Ljusternik-Schnirelman eigenvalues?
Is the set of all eigenvalues of the p-Laplacian with p6= 2discrete when N2?
Is every eigenvalue of nite multiplicity?
We refer to the work by P. Lindqvist [81] for more information regarding the eigenvalue problem (4.2).
4.4 The uniqueness of the limit of the sequence of principal eigen-
functions of p-Laplacian as p!1
For eachp2(1;1), one can show the existence of a principal eigenvalue of problem (4.2), 1(p), i.e.
the smallest of all possible eigenvalues , which can be characterized from a variational point of view

4. Further directions of research 86
in the following manner
1(p) := inf
u2C1
0(
)nf0gZ

jrujpdx
Z

jujpdx:
Moreover, its corresponding (principal) eigenfunctions are minimizers of 1(p) that do not change sign
in
. If for each p2(1;1), we letup>0 be an eigenfunction corresponding to the eigenvalue 1(p),
then there exists a subsequence of fupgwhich converges uniformly in
, when p!1 , to a nontrivial
and nonnegative solution, dened in the viscosity sense, of the limiting problem
8
><
>:minfjruj1u;
1ug= 0 in
u= 0 on @
;(4.3)
where
1u:=hD2uru;ruistands for the1-Laplace operator and
1:=1
max
x2
dist(x;@
)
(see, Juutinen, Lindqvist & Manfredi [68] or Fukagai, Ito & Narukawa [54]). Note that dist(
;@
) is
not always a viscosity solution of (4.3), but, in the particular case when
is a ball it turns out that
dist(
;@
) is the only viscosity solution of (4.3). However, for general domains
the convergence
of the entire sequence upto a unique limit, as p!1, is an open question .

Bibliography
[1] R. Adams: Sobolev Spaces , Academic Press, New York, 1975.
[2] W. Allegretto: Form estimates for the p(x)-Laplacean, Proc. Amer. Math. Soc. 135(2007), 2177-2185.
[3] J. C. Alvarez Paiva, C. Dur an: An Introduction to Finsler Geometry , Notas de la Escuela Venezolana de Mat ematicas,
1998.
[4] A. Alvino, V. Ferone, G. Trombetti & P.-L. Lions: Convex symmetrization and applications, Ann. Inst. Henri
Poincar e Anal. Non Lin eaire 14(1997), 275-293.
[5] S. Antontsev, J. I. D az & S. Shmar ev: Energy methods for free boundary problems: Applications to nonlinear PDEs
and
uid mechanics , Progress in Nonlinear Dierential Equations and their Applications, vol. 48, Birkh auser Boston
Inc., Boston, 2002.
[6] H. Attouch, G. Buttazzo, & G. Michaille: Variational Analysis in Sobolev and BV Spaces , in MPS-SIAM Series on
Optimization: SIAM and MPS, Philadelphia, 2006.
[7] D. Averna, S. Tersian, E. Tornator: On the existence and multiplicity of solutions for Dirichlet's problem for fractional
diferential equations, Fract. Calc. Appl. Anal. 19(2016), 253-266.
[8] M. Badiale, E. Serra: Semilinear Elliptic Equations for Beginners. Existence Results via the Variational Approach ,
Springer, 2011.
[9] G. Bellettini, M. Paolini: Anisotropic motion by mean curvature in the context of Finsler geometry, Hokkaido Math.
J.25(1996), 537-566.
[10] M. Belloni, V. Ferone & B. Kawohl: Isoperimetric inequalities, Wul shape and related questions for strongly
nonlinear elliptic operators, J. Appl. Math. Phys. (ZAMP) 54(2003), 771-783.
[11] M. Belloni, B. Kawohl: The pseudo- p-Laplace eigenvalue problem and viscosity solutions as p!1 ,ESAIM Control
Optim. Calc. Var. 10(2004), 28-52.
[12] M. Belloni, B. Kawohl & P. Juutinen: The p-Laplace eigenvalue problem as p!1 in a Finsler metric, J. Eur.
Math. Soc. 8(2006), 123-138.
[13] M. Bendahmane, K. H. Karlsen: Renormalized solutions of an anisotropic reaction-diusion-advection system with
L1data, Commun. Pure Appl. Anal. 5(2006), 733{762.
[14] M. Bendahmane, M. Langlais & M. Saad: On some anisotropic reaction-diusion systems with L1-data modeling
the propagation of an epidemic disease, Nonlinear Anal. 54(2003), 617{636.
[15] J. Bertoin: Levy Processes , vol. 121, Cambridge University Press, Cambridge, 1996.
[16] T. Bhattacharya, E. DiBenedetto & J. Manfredi: Limits as p!1 of
pup=fand related extremal problems,
Rend. Sem. Mat. Univ. Politec. Torino , special issue (1991), 15-68.
[17] M. Bocea, M. Mih ailescu: Eigenvalue problems in Orlicz-Sobolev spaces for rapidly growing operators in divergence
form, J. Dierential Equations 256(2014), 640-657.
87

Bibliography 88
[18] M. Bocea, M. Mih ailescu: Existence of nonnegative viscosity solutions for a class of problems involving the 1-
Laplacian, Nonlinear Dierential Equations and Applications (NoDEA) 23(2016), Article Number: 11.
[19] M. Bocea, M. Mih ailescu: On a family of inhomogeneous torsional creep problems, Proc. Amer. Math. Soc. 145
(2017), 4397-4409.
[20] M. Bocea, M. Mih ailescu & D. Stancu-Dumitru: The limiting behavior of solutions to inhomogeneous eigenvalue
problems in Orlicz-Sobolev spaces, Adv. Nonlinear Stud. 14(2014), 977{990.
[21] H. Brezis: Functional Analysis, Sobolev Spaces and Partial Dierential Equations , Springer, 2011.
[22] A. Braides: G-convergence for beginners , Oxford University Press, 2002.
[23] L. Brasco, E. Parini & M. Squassina: Stability of variational eigenvalues for the fractional p-Laplacian, Discrete and
Continuous Dynamical Systems 36(2016), 1813-1845.
[24] L. Caarelli: Nonlocal equations, drifts and games , in: Nonlinear Partial Dierential Equations, vol. 7, (2012), pp
37-52.
[25] P. Caldiroli, R. Musina: On the existence of extremal functions for a weighted Sobolev embedding with critical
exponent, Calc. Var. 8(1999), 365-387.
[26] P. Caldiroli, R. Musina: On a variational degenerate elliptic problem, Nonlinear Dier. Equ. Appl. (NoDEA) 7
(2000), 187-199.
[27] F. Charro, E. Parini: Limits as p!1 ofp-Laplacian problems with a superdiusive power-type nonlinearity:
Positive and sign-changing solutions, J. Math. Anal. Appl. 372(2010), 629{644.
[28] M. Chipot: Elliptic Equations: An Introductory Course , BAT – Birkh auser Advanced Texts, Birkh auser Verlag,
Basel-Boston-Berlin, 2009.
[29] P. Cl ement, B. de Pagter, G. Sweers & F. de T elin: Existence of solutions to a semilinear elliptic system through
Orlicz-Sobolev spaces. Mediterr. J. Math. 1(2004), 241{267.
[30] M. Cozzi, A. Farina & E. Valdinoci: Monotonicity formulae and classication results for singular, degenerate,
anisotropic PDEs, Adv. Math. 293(2016), 343{381.
[31] G. Dal Maso: An introduction to -convergence. Progress in nonlinear dierential equations and their applications,
vol. 8. Boston, MA: Birk auser, 1993.
[32] R. Dautray, J.-L. Lions: Mathematical analysis and numerical methods for science and technology, Vol. 1: physical
origins and classical methods , Springer, Berlin, 1985.
[33] E. De Giorgi: Sulla convergenza di alcune succesioni di integrali del tipo dell'area. Rend. Mat. 8(1975), 277-294.
[34] E. De Giorgi, T. Franzoni: Su un tipo di convergenza variazionale. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat.
Natur. 58(1975), 842{850.
[35] L. Del Pezzo, J. Fernandez Bonder & L. Lopez Rios: An optimization problem for the rst eigenvalue of the p-
fractional Laplacian , preprint arXiv:1601.03019v1 .
[36] L. Del Pezzo, A. Quaas: Global bifurcation for fractional p-Laplacian and an application, Z. Anal. Anwend. 35
(2016), 411{447.
[37] L. Del Pezzo, J. D. Rossi: Eigenvalues for a nonlocal pseudo p-Laplacian, Discrete and Continuous Dynamical
Systems 36(2016), 6737{6765.
[38] A. Di Castro, E. Montefusco: Nonlinear eigenvalues for anisotropic quasilinear degenerate elliptic equations, Non-
linear Analysis 70(2009), 4093{4105.

Bibliography 89
[39] A. Di Castro, M. P erez-Llanos & J. M. Urbano: Limits of anisotropic and degenerate elliptic problems, Comm. Pure
Appl. Anal. 11(2012), 1217{1229.
[40] E. Di Nezza, G. Palatucci & E. Valdinoci: Hitchhiker's guide to the fractional Sobolev spaces, Bull. Sci. Math. 136
(2012), 521-573.
[41] T. Donaldson: Nonlinear elliptic boundary value problems in Orlicz- Sobolev spaces, J. Di. Equations 10(1971),
507-528.
[42] X. Fan, Q. Zhang & D. Zhao: Eigenvalues of p(x)-Laplacian Dirichlet problem, Journal of Mathematical Analysis
and Applications 302(2005), 306-317.
[43] M. F arc a seanu: An eigenvalue problem involving an anisotropic dierential operator, Complex Variables and Elliptic
Equations ,62(2017), 297-306.
[44] M. F arc a seanu: On an eigenvalue problem involving the fractional ( s;p)-Laplacian, Fract. Calc. Appl. Anal. ,21
(2018), 94-103.
[45] M. F arc a seanu, M. Mih ailescu: Continuity of the rst eigenvalue for a family of degenerate eigenvalue problems,
Arch. Math. ,107(2016), 659-667.
[46] M. F arc a seanu, M. Mih ailescu: On a family of torsional creep problems involving rapidly growing operators in
divergence form, Proceedings of the Royal Society of Edinburgh Section A: Mathematics , in press.
[47] M. F arc a seanu, M. Mih ailescu & D. Stancu-Dumitru: A maximum principle for a class of rst order dierential
operators, New Trends in Dierential Equations, Control Theory and Optimisation, pp. 93-103, World Sci. Publ.,
Hackensack, NJ, (2016).
[48] M. F arc a seanu, M. Mih ailescu & D. Stancu-Dumitru: On the set of eigenvalues of some PDEs with homogeneous
Neumann boundary condition, Nonlinear Analysis 116(2015), 19-25.
[49] M. F arc a seanu, M. Mih ailescu & D. Stancu-Dumitru: Perturbed fractional eigenvalue problems, Discrete and Con-
tinuous Dynamical Systems-A ,37(2017), 6243-6255.
[50] M. F arc a seanu, M. Mih ailescu & D. Stancu-Dumitru: On the convergence of the sequence of solutions for a family
of eigenvalue problems, Mathematical Methods in the Applied Sciences ,40(2017), 6919-6926.
[51] M. F arc a seanu, M. Mih ailescu & D. Stancu-Dumitru: On a family of torsional creep problems in Finsler metrics,
submitted.
[52] I. Fragala, F. Gazzola & B. Kawohl: Existence and nonexistence results for anisotropic quasilinear elliptic equations,
Ann. I. H. Poincare-AN 21(2004), 715-734.
[53] G. Franzina, G. Palatucci: Fractional p{eigenvalues, Riv. Mat. Univ. Parma 5(2014), 373-386.
[54] N. Fukagai, M. Ito & K. Narukawa: Limit as p!1 ofp-Laplace eigenvalue problems and L1-inequality of Poincar e
type, Dierential Integral Equations 12(1999), 183-206.
[55] N. Fukagai, M. Ito & K. Narukawa: Positive solutions of quasilinear elliptic equations with critical Orlicz-Sobolev
nonlinearity on RN,Funkcial. Ekvac.. 49(2006), 235-267.
[56] J. P. Garcia Azorero, I. Peral Alonso: Existence and nonuniqueness for the p-Laplacian: nonlinear eigenvalues,
Comm. Partial Dierential Equations 12(1987) 1389-1430.
[57] M. Garc a-Huidobro, V. K. Le, R. Man asevich & K. Schmitt: On principal eigenvalues for quasilinear elliptic
dierential operators: an Orlicz-Sobolev space setting, Nonlinear Dierential Equations Appl. (NoDEA) 6(1999),
207{225.
[58] D. Gilbarg, N. S. Trudinger: Elliptic partial dierential equations of second order , Springer, Berlin, 1998.

Bibliography 90
[59] G. Gilboa, S. Osher: Nonlocal operators with applications to image processing, Multiscale. Model. Simul. 7(2008),
1005-1028.
[60] C. Gordon, L. Webb & S. Wolpert: One cannot hear the shape of a drum, Bull. Amer. Math. Soc. 27(1992), 134-138.
[61] J. P. Gossez: A strongly nonlinear elliptic problem in Orlicz-Sobolev spaces, Proc. Symp. Pure Mathematics 45
(1986), 455-462.
[62] J. P. Gossez: Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coecients,
Trans. Amer. Math. Soc. 190(1974), 163-205.
[63] P. Grisvard: Elliptic problems in nonsmooth domains , Pitman, Boston, MA, 1985.
[64] T. Ishibashi, S. Koike: On fully nonlinear PDE's derived from variational problems of Lpnorms, SIAM J. Math.
Anal. 33(2001), 545{569.
[65] R. Jensen: Uniqueness of Lipschitz extensions: Minimizing the up norm of the gradient, Arch. Rational Mech. Anal.
123(1993), 51-74.
[66] J. Jost, X. Li-Jost: Calculus of Variations , Cambridge University Press, 2008.
[67] P. Juutinen: Minimization problems for Lipschitz functions via viscosity solutions, Thesis University of Jyvaskyla
(1996), 1-39.
[68] P. Juutinen, P. Lindqvist & J. J. Manfredi: The 1-eigenvalue problem, Arch. Rational Mech. Anal. 148 (1999),
89-105.
[69] M. Kac: Can one hear the shape of a drum?, Amer. Math. Monthly 73(1966), 1-23.
[70] L. M. Kachanov: The theory of creep , Nat. Lending Lib. for Science and Technology, Boston Spa, Yorkshire, England,
1967.
[71] L. M. Kachanov: Foundations of the theory of plasticity , North Holland, Amsterdam-London, 1971.
[72] B. Kawohl: On a family of torsional creep problems: J. Reine Angew. Math. 410(1990), 1-22.
[73] M. Krasnosels'kii, J. Rutic'kii: Convex Functions and Orlicz Spaces , Noordho, Groningen, 1961.
[74] N. Laskin: Fractional quantum mechanics and Levy path integrals, Phys. Lett. A 268(2000), 298-305.
[75] A. L^ e: Eigenvalue problems for the p-Laplacian, Nonlinear Anal. 64(2006), 1057{1099.
[76] V. K. Le, K. Schmitt: Quasilinear elliptic equations and inequalities with rapidly growing coecients, J. London
Math. Soc. 62(2000), 852-872.
[77] G. M. Lieberman: On the regularity of the minimizer of a functional with exponential growth, Comment. Math.
Univ. Carolinae 33(1992), 45-49.
[78] E. Lindgren, P. Lindqvist: Fractional eigenvalues, Calc. Var. 49(2014), 795-826.
[79] P. Lindqvist: On the equation div( jrujp2ru) +jujp2u= 0, Proc. Amer. Math. Soc 109(1990), 157-164.
[80] P. Lindqvist: On non-linear Rayleigh quotients, Potential Analysis 2(1993), 199-218.
[81] P. Lindqvist: A nonlinear eigenvalue problem . Topics in mathematical analysis, 175-203, Ser. Anal. Appl. Comput.,
3, World Sci. Publ., Hackensack, NJ, 2008.
[82] P. Lindqvist, J. Manfredi: Note on 1-superharmonic functions, Revista Mathem atica de la Universidad Complutense
de Madrid 10(1997), 471{480.
[83] Y. Y. Li, L. Nirenberg: The distance function to the boundary, Finsler geometry, and the singular set of viscosity
solutions of some Hamilton-Jacobi equations, Comm. Pure Appl. Math. 58(2005), 85-146.

Bibliography 91
[84] M. Mih ailescu: An eigenvalue problem possessing a continuous family of eigenvalues plus an isolated eigenvalue,
Communications on Pure and Applied Analysis 10(2011), 701{708.
[85] M. Mih ailescu, G. Moro sanu: Eigenvalues of
p
qunder Neumann boundary condition, Canadian Mathematical
Bulletin 59(2016), 606-616.
[86] M. Mih ailescu, G. Moro sanu & V. R adulescu: Eigenvalue problems for anisotropic elliptic equations: an Orlicz-
Sobolev space setting, Nonlinear Analysis 73(2010), 3239-3252.
[87] M. Mih ailescu, G. Moro sanu & D. Stancu-Dumitru: Equations involving a variable exponent Grushin-type operator,
Nonlinearity 24(2011), 2663-2680.
[88] M. Mih ailescu, P. Pucci & V. R adulescu: Eigenvalue problems for anisotropic quasilinear elliptic equations with
variable exponent, Journal of Mathematical Analysis and Applications 340(2008), 687-698.
[89] M. Mih ailescu, V. R adulescu: Sublinear eigenvalue problems associated to the Laplace operator revisited, Israel
Journal of Mathematics 181(2011), 317-326.
[90] M. Mih ailescu, V. R adulescu & D. Stancu-Dumitru: On a Caarelli-Kohn-Nirenberg type inequality in bounded
domains involving variable exponent growth conditions and applications to PDE's, Complex Variables-Elliptic Equa-
tions 56(2011), 659-669.
[91] M. Mih ailescu, D. Stancu-Dumitru: A perturbed eigenvalue problem on general domains, Annals of Functional
Analysis ,7(2016), 529-542.
[92] L. E. Payne, G. A. Philippin: Some applications of the maximum principle in the problem of torsional creep, SIAM
J. Appl. Math. 33(1977), 446-455.
[93] M. P erez-Llanos, J. D. Rossi: The limits as p(x)!1 of solutions to the inhomogeneous Dirichlet problem of the
p(x)-Laplacian, Nonlinear Analysis 73(2010), 2027-2035.
[94] M. P erez-Llanos, J. D. Rossi: An anisotropic innity Laplacian obtained as the limit of the anisotropic ( p;q)-
Laplacian, Commun. Contemp. Math. 13(2011), 1057-1076.
[95] M. H. Protter, H. F. Weinberger: Maximum Principles in Dierential Equations , Englewood Clis, N.J., Prentice-
Hall, 1967.
[96] P. Pucci, V. R adulescu: Remarks on a polyharmonic eigenvalue problem, C. R. Acad. Sci. Paris 348(2010), 161-164.
[97] P. Pucci, J. Serrin: The maximum principle , Berlin: Birkh auser, 2007.
[98] M. M. Rao, Z. D. Ren: Theory of Orlicz Spaces , Marcel Dekker, Inc., New York, 1991.
[99] M. Struwe: Variational Methods: Applications to Nonlinear Partial Dierential Equations and Hamiltonian Systems ,
Springer, Heidelberg, 1996.
[100] A. Szulkin: Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value
problems, Ann. Inst. H. Poincar e Anal. Non Lin eaire 3(1986), 77-109.
[101] A. Szulkin, T. Weth: The method of Nehari manifold. Handbook of nonconvex analysis and applications , 597{632,
Int. Press, Somerville, MA, 2010.
[102] A. S. Tersenov, A. S. Tersenov: The problem of Dirichlet for anisotropic quasilinear degenerate elliptic equations,
J. Dierential Equations 235(2007), 376{396.
[103] M. Troisi: Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche di Mat. 18(1969), 3-24.
[104] N. S. Trudinger: On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure
Appl. Math. 20(1967), 721{747.

Bibliography 92
[105] J. V etois: Asymptotic stability, convexity, and Lipschitz regularity of domains in the anisotropic regime Communi-
cations in Contemporary Mathematics 12(2010), 35{53.
[106] M. Willem: Functional Analysis. Fundamentals and Applications , Birk auser, New York, 2013.
[107] Q. M. Zhou, K. Q. Wang: Existence and multiplicity of solutions for nonlinear elliptic problems with the fractional
Laplacian, Fract. Calc. Appl. Anal. 18(2015), 133{145.