Gradient Amzoiu (1) [615142]

LARGE SOLUTIONS FOR A CLASS OF
EQUATIONS WITH GRADIENT TERM
Manuel Amzoiu1, Roxana Sandulovici2
1University of Medicine and Pharmacy of Craiova, Faculty of Pharmacy,
200349 Craiova, Romania.
2Titu Maiorescu University, Faculty of Pharmacy, Department of
Pharmaceutical Technology, 040317 Bucharest, Romania.
Abstract
The main feature of this paper is the study of the quasilinear elliptic
equation with gradient term
pu+q(x)jrujp1=m(x)f(u). In this
equation let fbe a non-decreasing function with f >0 on (0 ;1),f(0) = 0
andq; m be a non-negative continuous functions.The aim is to nd how
the gradient term can in
uence the existence of solutions or its asymptotic
behavior.
Keywords: Explosive solution, elliptic equation, maximum principle
1 Introduction
In this paper we study the next elliptic equation with gradient term
8
<
:
pu+q(x)jrujp1=m(x)f(u) in
u0 in
(1)
where
2RN(N3) is a smooth either bounded or unbounded domain with
a compact (possibly empty )boundary. Throughout this paper we assume that
pis a real number with 1 < p <1,mandqare non-negative functions with
m;q2C0;(
) if
is bounded, and m;q2C0;
loc(
) if
is unbounded. The
non-decreasing non-linearity ffullls
(f1)f2C1[0;1);f00;f(0) = 0 and f >0 in (0;1)
(f2)Z1
1[F(t)]1
p<1whereF(t) =Zt
0f(s)ds
(f3)f(x)
(x+)p1is non-decreasing, for some 2R
1

We call the solution uof the problem (1) large (explosive; blowup) if
u(x)! 1 as dist(x;@
)!0 in the case that
is bounded. In the case

=RNuis called an entire large (explosive ) solution and the condition
becomesu(x)!1 asjxj!1 .
As we see from the remark 1.1 in [2] the equation has been studied for
p= 2 and without the gradient term in [5] where Carstea and Radulescu found
a necessary and sucient condition for the existence of a classical solution and
prove that this solution is large. The results obtained in this article extend the
results in [18] but in weaker condition.
In [6] Carstea and Radulescu extend the range of the equation by adding an
absorbtion term and in [7] a new way for proving the uniqueness is developed.
The gradient term appears in [14] and [16] and its in
uence on asymptotic
behavior is studied. The results obtained in the two articles were helpful in
solving semilinear elliptic systems in [15].
However going to
pis not straight forward because the p-Laplacian op-
erator is not linear and most of the proofs in the case p= 2 are done using
that linearity. A major breakthrough is made in [8] where Covei extended the
technique from [7] and adapted that method to work in equations where
pis
present. Also a new maximum principle is proved in this article.
Using the results obtained in [5] and [8] a generalization of the problem
from [5] is achieved in [2] where most of the uniqueness is done by adapting the
technique developed by Covei.
This article tries to see if the introduction of the gradient term in
uence
the conditions in which the large solution exists.
2 Auxiliary results
Lemma 2.1 Let
be a bounded domain. Assume that f satises (f1),q2
C0;(
)andg:@
!(0;1)is continuous.Then the problem
8
>><
>>:
pu+q(x)jrujp1=m(x)f(u) in
u=g on@
u0 in
(2)
has a unique positive solution.
Proof . Let us observe that the function u+(x) =nis a super-solution for prob-
lem (2), when nis suciently large . For a subsolution we consider an auxiliary
problem:
8
>><
>>:
pu=m(x)f(u) in
u=g in@
u0 in
(3)
2

that has a unique solution v, which is positive( see the theorem 5.1 in the
appendix). Thus u=vis a subsolution of (3). This implies that the problem
has a positive solution u2C1;(
).
For the uniqueness we consider that the equation (2) has two solutions uand
v. It is sucient to show that uvor, equivalently, ln( u(x)+)ln(v(x)+),
for anyx2
. We assume the contrary. So we have
lim
jxj!@
(ln(u(x) +)ln(v(x) +)) = 0
and we deduce that
max(ln(u(x) +)ln(v(x) +)) on
exists and is positive. Let us denote this point x0. Atx0we have
r(ln(u(x) +)ln(v(x) +)) = 0;
so1
u(x0) +
ru(x0) =1
v(x0) +
rv(x0);
which implies
1
(u(x0) +)p2
jru(x0)jp2=1
(v(x0) +)p2
jrv(x0)jp2: (4)
Using the condition ( f3) we obtain
f(u(x0))
(u(x0) +)p1>f(v(x0))
(v(x0) +)p1:
The observation 0 
(ln(u(x0) +)ln(v(x0) +));leads to

(u(x0))
u(x0) +
v(x0)
v(x0) +:
And by (4) it follows that
1
(u(x0) +)p1
jru(x0)jp2
u(x0)
=1
(v(x0) +)p1
jrv(x0)jp2
v(x0): (5)
Since
jrln(u(x0) +)jp2=1
(u(x0) +)p2
jru(x0)jp2;
it results that
r(jrln(u(x0) +)jp2) =
3

=(p2)jru(x0)jp2(u(x0) +)p3
(u(x0) +)2(p2)
ru(x0) +r(jru(x0)jp2)
(u(x0) +)p2:
The next conclusion emerges
r(jrln(u(x0) +)jp2)
rln(u(x0) +) =
=(p2)jru(x0)jp2jru(x0)j2
(u(x0) +)p+r(jru(x0)jp2)
ru(x0)
(u(x0) +)p1(6)
and
jrln(u(x0) +)jp2

ln(u(x0) +) =jru(x0)jp2
u(x0)
(u(x0) +)p1jru(x0)jp
(u(x0) +)p:
By (4), (5) and (6) we have
0
pln(u(x0) +)
pln(v(x0) +)
=
pu(x0)
(u(x0) +)p1(p1)jru(x0)jp
(u(x0) +)p
pv(x0)
(v(x0) +)p1+(p1)jrv(x0)jp
(v(x0) +)p
=
pu(x0)
(u(x0) +)p1
pv(x0)
(v(x0) +)p1=
=m(x0)f(u(x0))q(x0)jru(x0)jp1
(u(x0) +)p1m(x0)f(v(x0))q(x0)jru(x0)jp1
(v(x0) +)p1=
=m(x0)(f(u(x0))
(u(x0) +)p1f(v(x0))
(v(x0) +)p1)>0
and that is a contradiction. Hence uv. By symmetry we also obtain vu
and the proof of the uniqueness is now complete.
Another auxiliary result is needed.
Lemma 2.2 If the condition (f1) and (f2) are fullled, then
Z1
11
f(t)1
p1<1
The proof of this lemma is given in [2].
3 The main results
Theorem 3.1 Consider
to be a bounded domain and msatises
(m1)for every x 02
with q (x0) = 0; thereexist a domain
0which contains
x0such that
0
and q> 0on @
0.
Then the problem (1.1) has a positive large solution .
4

Theorem 3.2 Assume that the problem (1.1) has at least one solution for
=
RN. Ifmsatises the condition
(m1)' there exists a sequence of smooth bounded domains (
n)n1such that

n
n+1,RN=S1
n=1
nand (m1) holds in
n, for every n1,
then a maximal solution U of (1.1) exists.
If m satises the additional condition
(m2)R1
0r(r)dr<1where (r) =maxjxj=rq(x)
then U is an entire large solution.
Theorem 3.3 If the problem (1.1) has at least a solution for an unbounded

6=RNand m satises (m1)', then there exists a maximal solution U for
the problem (1.1). If m satises (m2), with (r) = 0 forr2[0;R]and
=
RNnB(0;R), then U is a large solution that blows-up at innity.
4 Proof of Theorem 3.1
Using lemma (2.1), the boundary value problem
8
>><
>>:
pvn+q(x)jrvnjp1= (m(x) +1
n)f(vn) in
vn=n on@
vn0 in
(7)
has a unique positive solution, for any n1.
We rst observe that the sequence vnis non-decreasing. Using lemma (2.1) the
problem
8
>><
>>:
p+q(x)jrvnjp1= (jjmjj1+ 1)f() in
= 1 on @
 >0 in
(8)
has a unique solution. Then we obtain with the maximum principle
0<v1v2:::vn:::in
: (9)
We claim that
(a) for allx02
there exists an open set #
containing x0andM0=
m0(x0)>0 such that vnM0in#, for anyn1;
(b)limx!@
v(x) =1, wherev(x) = limn!1vn(x):
Also we observe that (a) and (b) are sucient to complete the proof. From
(a) we obtain that the sequence ( vn) is uniformly bounded on every compact
subsets of
. Then, using (9) and (b), we prove that vis a large solution.
5

To prove (a) we distinguish two cases:
Caseq(x0)>0: By the continuity of q, there exists a ball B=B(x0;r)
such that
m0:= min
x2Bq(x)>0:
Letwbe a positive solution of the problem
8
<
:
pw+q(x)jrwnjp1=m0f(w) in
w(x)!1 asx!@
(10)
By the lemma 2.1 it follows that vnwinB. Furthermore, wis bounded in
B(x0;r
2). We denote M0= sup#w, where#=B(x0;r
2) and we obtain (a).
Caseq(x0) = 0: The boundedness of
and (q1) implies there exists a domain
#
, which contains x0such thatq >0 on@#. Then for any x2@#there
exists a ball B(x;r)
and a constant Mx>0 such that vnMxonB(0;rx
2),
for anyn. But@#is compact and it can be covered with a nite number of
balls,B(xi;rxi
2); i= 1;:::;k 0. TakingM0= max(Mx1;:::;Mxk0) and applying
the maximum principle we obtain vnM0and (a) follows.
We now consider the problem
8
>><
>>:
pz=m(x) in
z= 0 on @
z0 in
(11)
that has a unique positive solution(by the maximum principle). To prove (b) it
is sucient to show
Z1
v(x)dt
f(t)1
p1z(x);for anyx2
: (12)
By Lemma 2.1 the left side of (12) is well dened in
.
For a easier following of the proof of (12) we denote u=R1
vn(x)f(t)1
p1dt
andv=z(x). We want to show that uvor, equivalently, ln( u(x) +)
ln(v(x) +), for anyx2
. We assume the contrary. So we have
lim
jxj!@
(ln(u(x) +)ln(v(x) +)) = 0
and we deduce that
max

(ln(u(x) +)ln(v(x) +))
exists and is positive. Let us denote this point x0. Atx0we have
r(ln(u(x) +)ln(v(x) +)) = 0;
6

so1
u(x0) +
ru(x0) =1
v(x0) +
rv(x0)
which implies
1
(u(x0) +)p2
jru(x0)jp2=1
(v(x0) +)p2
jrv(x0)jp2: (13)
The condition ( f3) yields to
f(u(x0))
(u(x0) +)p1>f(v(x0))
(v(x0) +)p1:
Following the same thinking as in theorem 1.1 for the uniqueness, and taking
into account that

pu=div(rvnjrvnjp2) =div(1
f(vn)
rvnjrvnjp2) =
f0(vn)
f2(vn)
jrvnj2jrvnjp21
f(vn)

pvn
we have
0
pln(u(x0) +)
pln(v(x0) +)
=
pu(x0)
(u(x0) +)p1(p1)jru(x0)jp
(u(x0) +)p
pv(x0)
(v(x0) +)p1+(p1)jrv(x0)jp
(v(x0) +)p
=
pu(x0)
(u(x0) +)p1
pv(x0)
(v(x0) +)p1=
=f0(vn(x0))
f2(vn(x0))
jrvn(x0)j2jrvn(x0)jp21
f(vn(x0))

pvn(x0)
(u(x0) +)p1
pz(x0)
(v(x0) +)p1>
>f0(vn(x0))
f2(vn(x0))
jrvn(x0)jp1
f(vn(x0))
m(x0)f(vn(x0)) +1
f(vn(x0))
q(x0)jrvn(x0)jp1
(u(x0) +)p1
+m(x0)
(u(x0) +)p1=
=f0(vn(x0))
f2(vn(x0))
jrvn(x0)jp+1
f(vn(x0))
q(x0)jrvn(x0)jp1
(u(x0) +)p1>0
and that is a contradiction. Hence the assumption is false and the proof is now
complete.
7

5 Proof of theorem 3.2
Let us consider the next boundary value problem
8
>><
>>:
pun+q(x)jrunjp1=m(x)f(un) in
n
un(x)!1 asx!@
n
un>0 in
(14)
Using Lemma 2.1 the problem considered has a solution. Since
n

;for all
n1, applying the maximum principle we obtain unun+1in
n. Since
RN=S1
n=1
nand
n
it follows that there exists n0=n0(x0) such that
x02
nfor allnn0andx02RN. We can dene U(x0) = limn!1un(x0).
As in [12] we obtain that U2C1;
loc(RN) and
pU=q(x)f(U).
To prove that Uis the maximal solution, let ube an arbitrary solution of (1.1).
By the maximum principle we obtain vnuin
n, for alln1. It follows
thatUuinRN.
We prove now that if msatises (m2), then Ublows-up at innity. For that is
sucient to nd w2C2;(RN) such that Uwandw(x)!1 asjxj!1:
let us consider now that wis the maximal solution of the problem
8
<
:
pw=m(x)f(w) in
w0 in
:(15)
To prove that Uwwe use the same argument as in theorem (3.1) with
u=Uandv=w. This completes our proof.
5.1 Proof of theorem 3.3
We use the same argument as in theorem 3.2, but with some changes due to the
fact that
6=RN.
Let (
n)n1be a sequence of domains given by the condition ( m1)0andvnthe
solution of the problem (14), for n1 xed. We know that unun+1in
n
and we set U(x) = limn!1vn(x). We nd that Uis a maximal solution to
(1.1). When
= RNnB(0;R), we suppose that (q2) is fullled with ( r) = 0
forr2[0;R]. To prove that Uis a maximal solution it is enough to show
that a positive function w2C(RNnB(0;R)) withUwinRNnB(0;R) and
w(x)!1 asjxj!1 and asjxj&R. As in theorem 3.2 wis the positive
large solution of the problem
8
<
:
pw=m(x)f(w) in
w0 in
:(16)
8

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10