Annali di Matematica Pura ed Applicata manuscript No. [619784]

Annali di Matematica Pura ed Applicata manuscript No.
(will be inserted by the editor)
A nonlinear elliptic eigenvalue-transmission problem with
Neumann boundary condition
Luminit ¸a Barbu
Gheorghe Moros ¸anu
Cornel
Pintea
Received: date / Accepted: date
Abstract LetWRN,N2, be a bounded domain which is divided into two sub-domains
W1andW2. Consider in Wan eigenvalue-transmission problem associated with the p-
Laplacian acting in W1and the q-Laplacian acting in W2, 1<p<q, with Dirichlet-Neumann
conditions on the interface separating the two sub-domains W1andW2(see (1.1) below). The
main result Theorem 2.1 states the existence of a sequence of eigenvalues for this eigenvalue
problem. The proof is based on the Ljusternik-Schnirelman principle. Using the method of
Lagrange multipliers for constrained minimization problems, we show (see Theorem 2.2)
that if 2p<qthen there exists an eigenfunction in any set of the form

u2W1;p(W);ujW22W1;q(W2);1
pZ
W1jujp+1
qZ
W2jujq=a
;a>0:
The case of Robin conditions on ¶Wand the Riemannian setting are also addressed.
Keywords Eigenvalues
transmission problem
Neumann boundary condition
Sobolev
space
Ljusternik-Schnirelman principle
Lagrange multipliers
Robin boundary
condition
Riemannian setting
Mathematics Subject Classification (2000) 35J47
35J50
35J57
Corresponding author
Luminit ¸a Barbu
Faculty of Mathematics and Computer Science
Ovidius University
124 Mamaia Blvd, 900527 Constant ¸a, Romania
E-mail: [anonimizat]
Gheorghe Moros ¸anu
1. Academy of Romanian Scientists
Splaiul Independent ¸ei, Nr. 54, Sector 5, 050094 Bucharest, Romania
2. Faculty of Mathematics and Computer Science
Babes ¸-Bolyai University
1 M. Kog ˘alniceanu Str., 400084 Cluj-Napoca, Romania
E-mail: [anonimizat]
Cornel Pintea
Faculty of Mathematics and Computer Science
Babes ¸-Bolyai University
1 M. Kog ˘alniceanu Str., 400084 Cluj-Napoca, Romania
E-mail: [anonimizat]

2
1 Introduction
Consider a bounded domain WRN;N2, with Lipschitz boundary ¶W, which is divided
into two Lipschitz sub-domains W1andW2by a Lipschitz closed hypersurface H. We further
assume that H\¶Wis an(N2)-dimensional manifold. In the differentiable category this
is the case whenever Hand¶Wintersect transversally. In other words, W=W1[W2[G,
where G=H\W:The standard example we have in mind is the disc DNdivided by some
coordinate hyperplane in two open components, i.e. the two open semidiscs. Deformations
of this divided disc is a good enough source of further examples. The boundary of Wis
assumed smooth enough and is divided into two pieces ¶W 1and¶W 2in such a way that ¶W 1
is the union G1[Gand¶W 2is the union G2[G. To this picture we consider the eigenvalue
problem8
>>>>>>><
>>>>>>>:Dpu1=lju1jp2u1inW1; (1)
Dqu2=lju2jq2u2inW2; (2)
¶u1
¶np=0 on G1;¶u2
¶nq=0 on G2;(3)
u1=u2;¶u1
¶np=¶u2
¶nqonG; (4)(1.1)
where Drstands for the r-Laplace operator, namely Drw:=div
jÑwjr2Ñw

and¶
¶nrdenotes the boundary operator defined by
¶w
¶nr:=jÑwjr2¶w
¶nforr=p;q;1<p<q:
The solution u= (u1;u2)of the problem (1.1) is understood in a weak sense, i.e., uis an
element of the space
W:=
u2W1;p(W):ujW22W1;q(W2)
; (1.2)
where ui=ujWisatisfies the nonlinear problem (1.1) ionWiin the sense of distributions,
i=1;2, and u1,u2satisfy the boundary and transmission conditions (1.1) 3;4in the sense of
traces. Recall that, for any domain ˆWRNwith Lipschitz boundary ¶ˆW, the trace operator
Tr:W1;p(ˆW)!W11=p;p(¶ˆW)
is a linear and bounded operator, for 1 p<¥(see Gagliardo [8]). For linear transmission
problems, involving the Laplace operator or some perturbed Stokes operators, treated by
using the layer potential technique, we refere the reader to [5] and [9] respectively.
Definition 1.1 A scalar l2Ris said to be an eigenvalue of the problem (1.1) whenever
(1.1) admits a nontrivial solution u= (u1;u2)2W. In that case u= (u1;u2)is called an
eigenfunction /eigencouple of the problem (1.1) (which corresponds to the eigevalue l) and
the pair (u;l)aneigenpair of the problem (1.1). Note that W1;q(W)is a subspace of W, as
W1;q(W)is a subspace of W1;p(W).
We endow Wwith the norm
kukW:=kujW1kW1;p(W1)+kujW2kW1;q(W2);8u2W;
wherek
kW1;p(W1)andk
kW1;q(W2)are the usual norms of the Sobolev spaces W1;p(W1)
andW1;q(W2)respectively.

3
Remark 1.1 The space Wdefined before can be identified with the space
eW:=f(u1;u2)2W1;p(W1)W1;q(W2); TrW1u1=TrW2u2onGg; (1.3)
which shows that Wis a reflexive Banach space, as ˜Wis a closed subspace of the reflexive
product W1;p(W1)W1;q(W2)with reflexive factors.
While the inclusion WeWis obvious, for the opposite one we consider (u1;u2)2eW
and define
u(x) =(
u1(x);x2W1;
u2(x);x2W2:
Let us show that u2W:Obviously ubelongs to Lp(W);and its (distributional) derivatives
verify the equalities:
D¶u
¶xi;jE
=D
u;¶j
¶xiE
=Z
W1[W2u¶j
¶xidx
=Z
W1¶u
¶xijdxZ
¶W1un1ijds+Z
W2¶u
¶xijdxZ
¶W2un2ijds;
for all j2C¥
0(W), where n1= (n11;:::;n1n)andn2= (n21;:::;n2n)are the outward point-
ing unit normal fields to boundaries ¶W 1and¶W 2respectively. Clearly the integral terms
on the two boundaries cancel each other as u1=u2andn1i+n2i=0;8i=1;n;onG. Thus
D¶u
¶xi;jE
=Z
W1¶u
¶xijdx+Z
W2¶u
¶xijdx;8j2C¥
0(W);
which shows that
¶u
¶xi
W1=¶u1
¶xiand¶u
¶xi
W2=¶u2
¶xi;
for all i=1;n, and the desired claim follows now easily.
Proposition 1.1 The scalar l2Ris an eigenvalue of the problem (1.1) if and only if there
exists u =ul2Wnf0gsuch that
Z
W1jÑujp2Ñu
Ñw dx+Z
W2jÑujq2Ñu
Ñw dx
=lZ
W1jujp2uw dx +Z
W2jujq2uw dx
;8w2W:(1.4)
Proof Indeed, if u2Wis a solution of the problem (1.1), then we have for all w2W
Z
W1div
jÑujp2Ñu

w dx+Z
W2div
jÑujq2Ñu

w dx
=lZ
W1jujp2uw dxlZ
W2jujq2uw dx
or, equivalently,
Z
W1jÑujp2Ñu
Ñw dx+Z
GwjÑujp2¶u
¶ndsZ
W2jÑujq2Ñu
Ñw dx
Z
GwjÑujq2¶u
¶nds=lZ
W1jujp2uw dxlZ
W2jujq2uw dx ;

4
which is equivalent to (1.4).
Conversely, assume that u2Wsatisfies (1.4) and consider w2Wsuch that wjW1=j
for some arbitrary j2C¥
0(W1)andwjW2=0:We obtain
Z
W1jÑujp2Ñu
Ñjdx=lZ
W1jujp2ujdx;8j2C¥
0(W1):
By using the formula of integration by parts, we obtain
Z
W1div
jÑujp2Ñu

jdx=lZ
W1jujp2ujdx;8j2C¥
0(W1);
which shows thatDpu=ljujp2uinW1:Similarly,Dqu=ljujq2uinW2:
We next assume that w2C1(W)andwjW2=0:With such a choice of w, using the
integration by parts formula, the fact that wjG=0 and the equation Dpu=ljujp2uin
W1obtained above, the relation (1.4) implies
0=Z
¶W1wjÑujp2¶u
¶nds=Z
G1wjÑujp2¶u
¶nds
for all w2C1(W1);wjG=0, therefore¶u
¶np=jÑujp2¶u
¶n=0 onG1:One can similarly
show that¶u
¶nq=jÑujq2¶u
¶n=0 onG2:
It remains to obtain the transmission conditions on G. First of all, it is obvious that
TrW1(ujW1) =TrW2(ujW2)onG. Finally, we take in (1.4) w=j, where jis an arbitrary
function in C¥
0(W). Using again the integration by parts formula (in particular, on Gwe have
n1+n2=0;the normal vector nkbeing chosen to point towards the exterior of Wk;k=1;2)
and the equations and equalities proved before we derive
Z
Gj¶u
¶npds+Z
Gj¶u
¶nqds=0;8j2C¥
0(W):
Thus, the transmission relation
¶u
¶np=¶u
¶nqonG
is satisfied. This completes the proof.
If we choose w=uin (1.4) we see that there exist no negative eigenvalues of problem
(1.1). It is also obvious that l0=0 is an eigenvalue of this problem and the corresponding
eigenfunctions are the nonzero constant functions. So any other eigenvalue belongs to (0;¥).
Obviously ucorresponding to any eigenvalue l>0 cannot be a constant function (see
(1.4) with w=u).
If we assume that l>0 is an eigenvalue of problem (1.1) and choose w1 in (1.4) we
deduce that every eigenfunction ucorresponding to lsatisfies the equation
Z
W1jujp2u dx+Z
W2jujq2u dx=0:
So all eigenfunctions corresponding to positive eigenvalues necessarily belong to the set
C:=n
u2W;Z
W1jujp2u dx+Z
W2jujq2u dx=0o
: (1.5)

5
Using the Sobolev’s embedding theorem and [11, Lemma A1]) we can see that Cis a weakly
closed subset of W. This set has nonzero elements. To show this, we choose x1;x22W1;x16=
x2,r>0, such that Br(x1)\Br(x2) =/ 0;Br(xk)W1, and consider the test functions uk:
W!R;k=1;2;
uk(x) =8
><
>:e1
r2jxxkj2;ifx2Br(xk);
0; otherwise.
Clearly uk2W,k=1;2. Denote
qk=Z
Wup1
kdx:
Obviously qk>0,k=1;2. Define sk=q1
p1
k;k=1;2. It is then easily seen that the function
w=s1u1s2u2belongs to Cnf0g.
Our next goal is to prove, via the Ljusternik-Schnirelman principle, that there exists a
sequence of positive eigenvalues of problem (1.1). Note however that this sequence might
not cover the whole eigenvalue set.
2 Results
In what follows we make use of a version of Ljusternik-Schnirelman principle (see [2], [19,
Section 44.5, Remark 44.23] and [11]) in order to establish the existence of a sequence of
eigenvalues for problem (1.1).
Define the functionals F;G:W!Rby
F(u):=1
pZ
W1jujpdx+1
qZ
W2jujqdx (2.1)
G(u):=1
pZ
W1
jÑujp+jujp

dx+1
qZ
W2
jÑujq+jujq

dx;
=F(u)+1
pZ
W1jÑujpdx+1
qZ
W2jÑujqdx: (2.2)
It is easily seen that functionals FandGare of class C1onW(see Remark 2.1 below) and
obviously F;Gare even with F(0) =G(0) =0. We also have
hF0(u);wi=Z
W1jujp2uw dx +Z
W2jujq2uw dx ; (2.3)
hG0(u);wi=hF0(u);wi+Z
W1jÑujp2Ñu
Ñw dx+Z
W2jÑujq2Ñu
Ñw dx; (2.4)
for all w2W. We denote by SG(1)the level set of G;SG(1):=fu2W;G(u) =1g:
We have the following auxiliary result:
Lemma 2.1 The functionals F and G satisfy the following properties:
(h1)F0is strongly continuous, i.e., u n*u (meaning u n!u weakly) in W)F0(un)!F0(u)
and
hF0(u);ui=0)u=0;

6
(h2)G0is bounded and satisfies condition (S0);i. e.,
un*u;G0(un)*w;hG0(un);uni!h w;ui) un!u;
(h3)SG(1)is bounded and if u 6=0then
hG0(u);ui>0;lim
t!¥G(tu) =¥;inf
u2SG(1)hG0(u);ui>0:
Proof (h1)Assume that un*uinW. H¨older’s inequality yields
jhF0(un)F0(u);wijkj unjp2unjujp2uk
Lp
p1(W1)kwkLp(W1)
+kjunjq2unjujq2uk
Lq
q1(W2)kwkLq(W2)

kjunjp2unjujp2uk
Lp
p1(W1)
+kjunjq2unjujq2uk
Lq
q1(W2)!
kwkW;(2.5)
for all w2W. This shows that the linear functionals F0(un)F0(u)are all bounded and
kF0(un)F0(u)kkj unjp2unjujp2uk
Lp
p1(W1)
+kjunjq2unjujq2uk
Lq
q1(W2); (2.6)
for all n1. Since un*uinWit follows thatfungas well as the sequences of restrictions
fun
W1gandfun
W2gare bounded (see [1, Proposition 3.5, p. 58]). Consequently un!uin
Lp(W),unjW1!ujW1inLp(W1)andunjW2!ujW2inLq(W2), as the canonical injections
W1;p(W),!Lp(W),W1;p(W1),!Lp(W1)andW1;q(W2),!Lq(W2)are all compact (see
[20, Proposition 21.29, p. 262]). The convergence kunkLp(W1)!k ukLp(W1)is equivalent
with Z
W1jjunjp2unjp
p1dx!Z
W1jjujp2ujp
p1dx: (2.7)
As the set of weak cluster points of the sequence (junjp2un)inLp=(p1)(W1)is the single-
tonfjujp2ug, it follows that in fact this sequence is strongly convergent in Lp=(p1)(W1)
tojujp2u(see, e.g., [1, Prop. 3.32, p. 78]).
One can similarly show that junjq2un!jujq2uinLq=(q1)(W2). Thus, the conver-
gence F0(un)!F0(u)inWfollows by using (2.6).
IfhF0(u);ui=0 then obviously u=0.
Note that the strong continuity of Gcan be similarly derived.
(h2)Let us first prove that for all u;w2Wthe following relations hold:
hG0(u)G0(w);uwi

ku1kp1
W1;p(W1)kw1kp1
W1;p(W1)

ku1kW1;p(W1)kw1kW1;p(W1)

(2.8)
+
ku2kq1
W1;q(W2)kw2kq1
W1;q(W2)

ku2kW1;q(W2)kw2kW1;q(W2)

0;

7
where u1;w1;u2;w2stand for u
W1;w
W1;u
W2;w
W2respectively. Moreover
hG0(u)G0(w);uwi=0,u=wa. e. in W: (2.9)
It is obvious that
hG0(u)G0(w);uwi=ku1kp
W1;p(W1)+kw1kp
W1;p(W1)
+ku2kq
W1;q(W2)+kw2kq
W1;q(W2)(T1+T2)(T3+T4);(2.10)
where we have denoted
T1:=Z
W1
jÑujp2Ñu
Ñw+jujp2uw

dx;
T2:=Z
W1
jÑwjp2Ñw
Ñu+jwjp2wu

dx;
andT3;T4are similarly defined, by replacing pandW1with qandW2. Using the H ¨older
inequality we obtain that
T1Z
W1jÑujpdxp1
pZ
W1jÑwjpdx1
p+Z
W1jujpdxp1
pZ
W1jwjpdx1
p
Z
W1(jÑujp+jujp)dxp1
pZ
W1(jÑwjp+jwjp)dx1
p(2.11)
=ku1kp1
W1;p(W1)kw1kW1;p(W1);
where we have also used the inequality
asg1s+bsd1s(a+b)s(g+d)1s;8a;b;g;d>0;s2(0;1):
Similar inequalities can be obtained for the other terms, T2;T3;T4;and using (2.10) we
derive (2.8).
Now by (2.8) we see that hG0(u)G0(w);uwi=0 implies
ku1kW1;p(W1)=kw1kW1;p(W1);ku2kW1;q(W2)=kw2kW1;q(W2); (2.12)
and also we have equalities in H ¨older inequalities therefore, there exist positive constants,
k1;k2such thatjuij=kijwij;i=1;2. On the other hand, we have equality in (2.11), thus
T1=kp1
1kw1kp
W1;p(W1))u1=k1w1a. e. in W1:
Similarly we can derive that u2=k2w2a. e. in W2and taking into account (2.12) we derive
(2.9).
In order to prove that G0is bounded we can use again the H ¨older inequality and straight-
forward computations lead us to
jhG0(u);wij
ku1kp1
W1;p(W1)+ku2kq1
W1;q(W2)

kwkW;8u;w2W:
Moreover, a similar argument to the one we used to prove (h1)would imply the continuity
ofG0.
Finally, let us prove that G0verifies condition (S0);i. e.,
un*u;G0(un)*w;hG0(un);uni!h w;uiimplies un!u;

8
for some u2W;w2W:Indeed, as un*uinW;we have unjW1!ujW1inLp(W1)and
unjW2!ujW2inLq(W2):Since Wis a reflexive Banach space, using the Lindenstrauss-
Asplund-Troyanski theorem (see [18]), it is enough to prove that kunkW!kukWin order
to obtain the strong convergence un!u:This convergence is a simple consequence of the
equality
lim
n!¥hG0(un)G0(u);unui=lim
n!¥
hG0(un);unihG0(un);uihG0(u);unui

=0
and the inequality (2.8).
The properties (h3)follow immediately from the definition of the functional G:Thus
the proof is complete.
Remark 2.1 For the convenience of the reader we recall that:
1. the C1-smooth regularity of the functionals FandGfollows by computing the G ˆateaux
deriavtives
hF0(u);wi=d
dt
t=0F(u+tw)andhG0(u);wi=d
dt
t=0G(u+tw)
ofFandGatu2Win the direction w2Wand showing that they have the forms (2.3)
and (2.4) respectively. The existence of the G ˆateaux derivatives of FandGat every
point of Wand all directions of Wcombined with the strong continuity of F0andG0,
shows the Fr ´echet differentiability of FandG, and therefore the C1-smooth regularity
ofFandG.
2. The weak closedness of the set Cdefined by (1.5) follows also from the strong continu-
ity of F0and the representation of Casfu2WjhF0(u);1i=0g.
Due to the properties (h1)(h3), verified by the functionals FandG, combined with
their properties to be even and to vanish at zero, it follows, according to the Ljusternik-
Schnirelman principle, that the eigenvalue problem
F0(u) =mG0(u);u2SG(1) (2.13)
admits a sequence of eigenpairs f(un;mn)gsuch that un*0 and mn!0 as n!¥and
mn6=0, for all n. In factfmngis a decreasing sequence of non-negative reals (which con-
verges to zero) and
mn=sup
H2Aninf
u2HF(u);8n2N; (2.14)
where Anis the class of all compact, symmetric subsets KofSG(1)such that F(u)>0 on
Kandg(K)n;where g(K)denotes the genus of K, i. e.,
g(K):=inffk2N;9h:K!Rknf0gsuch that h is continuous and odd g:
The problem (2.13) consists in finding u2SG(1)such that
Z
W1jujp2uw dx +Z
W2jujq2uw dx
=m
hF0(u);wi+Z
W1jÑujp2Ñu
Ñw dx+Z
W2jÑujq2Ñu
Ñw dx
;

9
for all w2W, or equivalently, in finding u2SG(1);such that
Z
W1jÑujp2Ñu
Ñw dx+Z
W2jÑujq2Ñu
Ñw dx
= (1=m1)Z
W1jujp2uw dx +Z
W2jujq2uw dx
;8w2W:(2.15)
Observe that (2.15) is the variational formulation of problem (1.1). We therefore get the
following consequence of the Ljusternik-Schnirelman principle associated with the trans-
mission problem (1.1):
Theorem 2.1 The sequencefmngof eigenvalues of the problem (2.13) produces a nonde-
creasing sequence ln=1
mn1of eigenvalues of the problem (1.1) and obviously ln!¥
as n!¥.
In what follows we shall use the Lagrange multipliers rule to show that every positive
level set of the functional Fdefined by (2.1) contains an eigenfunction of the problem (1.1)
and we shall find its corresponding eigenvalue in terms of the pointed out eigenfunction.
Such an eigenfunction will appear as a solution of the minimum problem
min
u2C\SF(a)H(u); (2.16)
where His defined by
H:W![0;¥);H(u):=1
pZ
W1jÑujpdx+1
qZ
W2jÑujqdx;8u2W; (2.17)
Cis defined by (1.5) and SF(a)is the set at the level a>0 ofF, i.e.
SF(a):=fu2W;F(u) =ag;8a>0:
In this respect we first recall the Lagrange multipliers principle (see, e.g., [14, Thm.
2.2.18, p. 78]):
Lemma 2.2 Let X ;Y be real Banach spaces and let f :D!Rbe Fr ´echet differentiable,
g2C1(D;Y), where DX is a nonempty open set. If v 0is a local minimizer of the constraint
problem
min f(w);g(w) =0;
andR(g0(v0))(the range of g0(v0)) is closed, then there exist l2Rand y2Y, at least
one of which is non zero, such that
lf0(v0)+yg0(v0) =0;
where Ystands for the dual of Y :
Note that l6=0 whenever g0(v0)is onto and can be therefore chosen to be 1 in this particular
case.
The eigenvalue problem corresponding to the minimmum problem (2.16), via the La-
grange multipliers, is:
H0(ua) =laF0(ua);la>0;ua6=0: (2.18)
Its variational version is (1.4).

10
Theorem 2.2 Let F and H be the functionals defined by (2.1) and(2.17) . For every 2p<
q;a>0, the minimization problem (2.16) has a solution u awhich is an eigenfunction of
the eigenvalue problem (2.18) and therefore a solution of the variational version (1.4) of the
initial eigenvalue problem (1.1) .
Proof Let us first show that the set C\SF(a)is nonempty for every a>0:Indeed, if we
choose w2C\C¥
0(W1), nonzero, then aw=F(w)2C\SF(a):
Now, the functional His coercive on the weakly closed subset C\SF(a)of the reflexive
Banach space W, i. e.,
lim
kukW!¥
u2C\SF(a)H(u) =¥:
This fact is a simple consequence of the equality
lim
kukW!¥
u2SF(a)(kÑukLp(W1)+kÑukLq(W2)) =¥:
On the other hand, the weakly lower semicontinuity of the norms in Lp(W1)andLq(W2)
implies the weakly lower semicontinuity of the functional HonC\SF(a):Then, we can
apply [16, Theorem 1.2] in order to obtain the existence of a global minimum point of Hover
C\SF(a), say ua, i. e., H(ua) =min u2C\SF(a)H(u). Obviously ua2C\SF(a)implies
thatuais a nonconstant function. In fact, uais a solution of the minimization problem
min
w2WH(w);
under the restrictions
g(w):=1
pZ
W1jwjpdx+1
qZ
W2jwjqdxa=0;
h(w):=Z
W1jwjp2w dx+Z
W2jwjq2w dx=0;8w2W:
We can apply Lemma 2.2 with X=W;D=Wnf0g;Y=R;f=H;g;h:W!Rbeing
the functions just defined above, and v0=ua;on the condition that R(g0(ua));R(h0(ua))
be closed sets. In fact we can show that g0(ua);h0(ua)are surjective, i.e.,8c1;c22Rthere
exist w1;w22Wsuch that
hg0(ua);w1i=c1;hh0(ua);w2i=c2:
We seek w1;w2of the form w1=bua;w2=g, with b;g2R. Thus we obtain from the
above equations
bZ
W1juajpdx+Z
W2juajqdx
=c1;
g
(p1)Z
W1juajp2dx+(q1)Z
W2juajq2dx
=c2
which have unique solutions b;gsince ua2SF(a)implies that
r1Z
W1juajp1dx+r2Z
W2juajq1dx>0;8p1;q1;ri>0;i=1;2:

11
Thus, by Lemma 2, there exist landm2Rsuch that, l2+m2>0 and for all w2W;
Z
W1jÑuajp2Ñua
Ñw dx+Z
W2jÑuajq2Ñua
Ñw dx
lZ
W1juajp2uaw dx+Z
W2juajq2uaw dx
m
(p1)Z
W1juajp2w dx+(q1)Z
W2juajq2w dx
=0:(2.19)
Testing with w=1 in (2.19) and observing that uabelongs to C, we deduce that m=0 and
therefore l6=0. By choosing w=uain (2.19) we find K1alK2a=0, where K1aandK2a
denote the constants
Z
W1jÑuajpdx+Z
W2jÑuajqdxandZ
W1juajpdx+Z
W2juajqdx
respectively, which are positive as ua2C\SF(a). In other words (2.19) becomes
Z
W1jÑuajp2Ñua
Ñw dx+Z
W2jÑuajq2Ñua
Ñw dx
=laZ
W1juajp2uaw dx+Z
W2juajq2uaw dx
;(2.20)
where
la=K1a
K2a=R
W1jÑuajpdx+R
W2jÑuajqdx
R
W1juajpdx+R
W2juajqdx:
Thus (la;ua)is an eigenpair of problem (1.4).
Remark 2.2 The results we have proved so far are also valid for the eigenvalue problem
obtained out of (1.1) by replacing the equation (1.1) 2with the equation
Dqu2=lju2jp2u2inW2; (2.21)
for 1 <p<q:In this case we shall consider the same space Wbut endowed with the norm
jkujk:=kukW1;p(W1)+kÑukLq(W2)+kukLp(W2);8u2W: (2.22)
Ifpq, thenjk
jk is a norm in Wequivalent with the usual norm k
k Wof this space. This
fact follows from [4, Proposition 3.9.55].
In this case, the variational version of the new eigenvalue problem is:
Findl2Rfor which there exists u2Wnf0gsuch that
Z
W1jÑujp2Ñu
Ñw dx+Z
W2jÑujq2Ñu
Ñw dx
=lZ
Wjujp2uw dx ;8w2W:(2.23)
In order to obtain the counterpart of Theorem 2.1 for this new eigenvalue transmission prob-
lem we need to verify the conditions (h1)(h3)of Lemma 2.1. We shall define for this new
context the corresponding functionals Fp;Gp:W![0;¥)
Fp(u):=1
pZ
Wjujpdx;Gp(u):=Fp(u)+1
pZ
W1jÑujpdx+1
qZ
W2jÑujqdx:(2.24)

12
All calculations are similar to those we did to prove (h1)(h3)in the case of the eigen-
value transmission problem (1.1), except the one which verifies the property (S0)onG0
pof
(h2):In order to prove (S0)we define the functional J:W!W
hJ(u);wi:=Z
W2jujp2uw dxZ
W2jujq2uw dx ;8u;w2W:
One can show, by using the same type of arguments as we did to prove (h1)and Lemma 2.1,
thatJ(u)is strongly continuous. Let us consider
un*u;G0
p(un)*wp;hG0
p(un);uni!h wp;uiasn!¥
for some u2W;wp2Wand we shall show that un!u:In this respect (see also the
argument within the proof of the statement (h2)) it is sufficient to show that kunkW!kukW,
ask
k Wandjk
jk are equivalent norms on W:In this regard we observe that
G0(un) =G0
p(un)J(un)*wpJ(u);hG0(un);uni!h wpJ(u);ui;
which combined with the (S0)property of G0implies the desired statement.
The counterpart of Theorem 2.2 can be obtained with no difficulty, by using arguments
similar to those we have used in the case of the eigenvalue transmission problem (1.1).
3 Extensions
In this section we discuss some extensions of the previous results.
An eigenvalue-transmission problem with Robin boundary conditions
Following the same type of arguments one can actually prove the counterparts of Theo-
rem 2.1 and Theorem 2.2 for the following more general eigenvalue-transmission problem,
involving Robin conditions on G1andG2, namely
8
>>>>>>>>>>><
>>>>>>>>>>>:Dpu1=lju1jp2u1inW1;
Dqu2=lju2jq2u2inW2;
¶u1
¶np+b1ju1jp2=0 on G1;
¶u2
¶nq+b2ju2jq2=0 on G2;
u1=u2;¶u1
¶np=¶u2
¶nqonG;(3.1)
where b1;b20. The variational version of problem (3.1) is:
Proposition 3.1 The scalar l2Ris an eigenvalue of the problem (3.1) if and only if there
exists u2Wnf0gsuch that
Z
W1jÑujp2Ñu
Ñw dx+Z
W2jÑujq2Ñu
Ñw dx
+b1Z
¶W1jujp2uw ds+b2Z
¶W2jujq2uw ds
=lZ
W1jujp2uw dx +Z
W2jujq2uw dx
;8w2W:(3.2)

13
While the functional playing the role of Fin this setting remains unchanged, the functional
playing the role of G:W!Ris given by
G(u):=1
pZ
W1
jÑujp+jujp

dx+1
qZ
W2
jÑujq+jujq

dx
+1
pZ
¶W1b1jujpds+1
qZ
¶W2b2jujqds:(3.3)
The counterpart of problem (1.1) in the Riemannian setting
Let(M;g)be a compact boundaryless Riemannian manifold and WMbe a connected
open set such that W:=MnWis also connected. We denote WbyW+and the common
boundary of W+andWby¶W, which is assumed to be a hypersurface of M. We consider
the following coupled problem
8
>>><
>>>:Dpu+=lju+jp2u+inW+;
Dqu=ljujq2uinW;
u+=u;¶u+
¶np=¶u
¶nqon¶W;(3.4)
where Drwstands for the r-Laplace operator div
jÑwjr2Ñw

.
Proposition 3.2 The scalar l2Ris an eigenvalue of the problem (3.4) if and only if there
exists u2WWnf0gsuch that
Z
W+jÑujp2Ñu
Ñw dx+Z
WjÑujq2Ñu
Ñw dx
=lZ
W+jujp2uw dx +Z
Wjujq2uw dx
;8w2WW;(3.5)
where
WW:=
u2W1;p(M):ujW2W1;q(W)
: (3.6)
The proof of Proposition 3.2 works along the same lines with the proof of Proposition 1.1
and partly relies on the intergation by parts formula [12, p. 383]
Z
X(fdivX)dVg=Z
Xg(X;grad f)dVg+Z
¶Xg(X;n)dV˜g
where (X;g)is a compact oriented Riemannian manifold, nis the outward unit normal
vector field on ¶Xand ˜gis the Riemannian metric on ¶Xinduced by g.
We endow WWwith the norm
kukWW:=kujW+kW1;p(W+)+kujWkW1;q(W);8u2WW
wherek
kW1;p(W+)andk
kW1;q(W)are the usual norms of the Sobolev spaces W1;p(W+)
andW1;q(W)respectively.
Remark 3.1 The space WWdefined before can be identified with the space
eWW:=f(u+;u)2W1;p(W+)W1;q(W); TrW+u+=TrWuon¶Wg:

14
Note that WWis a reflexive Banach space, as it is a closed subspace of the reflexive product
W1;p(W+)W1;q(W)with reflexive factors (see [1, p. 70], [6, p. 11] or [7, p. 20]). Define
the functionals FandGonWW:
F(u):=1
pZ
W+jujpdx+1
qZ
Wjujqdx; (3.7)
G(u):=1
pZ
W+
jÑujp+jujp

dx+1
qZ
W
jÑujq+jujq

dx; (3.8)
for all u2WW:It is easily seen that functionals FandGare of class C1onWWand obviously
F;Gare even with F(0) =G(0) =0. We also have
hF0(u);wi=Z
W+jujp2uw dx +Z
Wjujq2uw dx ;
hG0(u);wi=hF0(u);wi+Z
W+jÑujp2Ñu
Ñw dx+Z
WjÑujq2Ñu
Ñw dx;
for all w2WW:We denote by SG(1)the level setfu2WW;G(u) =1gofG.
Tthe following auxiliary result can be proved in a similar way with Lemma 2.1.
Lemma 3.1 The functionals F and G satisfy the following properties:
(h1)F0is strongly continuous, i.e., u n*u in W W)F0(un)!F0(u)and
hF0(u);ui=0)u=0;
(h2)G0is bounded and satisfies condition (S0);i. e.,
un*u;G0(un)*w;hG0(un);uni!h w;ui) un!u;
(h3)SG(1)is bounded and if u 6=0then
hG0(u);ui>0;lim
t!¥G(tu) =¥;inf
u2SG(1)hG0(u);ui>0:
According to the properties (h1)(h3), verified by the functionals FandG, combined with
their properties to be even and to vanish at zero, it follows, via the Ljusternik-Schnirelman
principle, that the eigenvalue problem
F0(u) =mG0(u);u2SG(1): (3.9)
admits a sequence of eigenpairs f(un;mn)gsuch that un*0,mn!0 asn!¥andmn6=0,
for all n.
Theorem 3.1 The sequencefmngof eigenvalues of the problem (3.9) produces a nonde-
creasing sequence ln=1
mn1of eigenvalues of the problem (3.4) and obviously ln!¥
as n!¥.

15
Consider the minimization problem
min
u2CW\SF(a)H(u); (3.10)
where
H:WW![0;¥);H(u):=1
pZ
W+jÑujpdx+1
qZ
WjÑujqdx;8u2WW; (3.11)
CW:=n
u2WW;Z
W+jujp2u dx+Z
Wjujq2u dx=0o
: (3.12)
andSF(a)is the set at the level a>0 ofF(i.e.SF(a):=fu2W;F(u) =ag).
The eigenvalue problem corresponding to the minimization problem (3.10), via the La-
grange multipliers, is:
H0(ua) =laF0(ua);la>0;ua6=0; (3.13)
Its variational version is (3.5).
Theorem 3.2 Let F and H be the functionals defined by (3.7) and(3.11) . For every 2p<
q;a>0, the problem (3.10) has a solution u awhich is an eigenfunction of the eigenvalue
problem (3.13) and therefore a solution of the variational version (3.5) of the eigenvalue
problem (3.4) .
Remark 3.2 Note that in (3.4) there is no boundary condition, because the ambient mani-
fold Mis boundary-free. One can think of the eigenvalue-transmission counterpart of the
problem (1.1) in a more general Riemannian setting, where the ambient manifold Mhas
nonempty boundary and the interface hypersurface is suitably chosen. Indeed, for a com-
pact Riemannian manifold (M;g)with nonempty boundary, we consider a connected open
setW1Msuch that W2:=MnW1is also connected and W1,W2are manifolds with bound-
aries¶W 1,¶W 2. We further assume that their common boundary part G:=¶W 1\¶W 2is
a hypersurface of M(which is closed), such that G1:=¶W 1nGandG2:=¶W 2nGare also
connected. With such choices the eigenvalue-transmission counterpart of the problem (1.1)
in this more general Riemannian setting, looks like (1.1).
Acknowledgements
The authors appreciate the referee’s comments and observations, as their implementation
improved the presentation.
Cornel Pintea was supported by a grant of the Romanian Ministry of Research and Inno-
vation, CNCS – UEFISCDI, project number PN-III-P4-ID-PCE-2016-0190, within PNCDI
III.
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