## Boundary Value Problems

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# Multiple positive solutions for semilinear elliptic systems involving subcritical nonlinearities in

Boundary Value Problems20122012:118

DOI: 10.1186/1687-2770-2012-118

Accepted: 4 October 2012

Published: 24 October 2012

## Abstract

In this paper, we investigate the effect of the coefficient of the subcritical nonlinearity. Under some assumptions, for sufficiently small , there are at least k (≥1) positive solutions of the semilinear elliptic systems

where , , for .

MSC:35J20, 35J25, 35J65.

### Keywords

semilinear elliptic systems subcritical exponents Nehari manifold

## 1 Introduction

For , , and , we consider the semilinear elliptic systems

where .

Let f, g and h satisfy the following conditions:

(A1) f is a positive continuous function in and .

(A2) there exist k points in such that

and .

(A3) where , and .

In [1], if Ω is a smooth and bounded domain in (), they considered the following system:
and proved the existence of a least energy solution in Ω for sufficiently small and . Lin and Wei also showed that this system has a least energy solution in for and . In this paper, we study the effect of of (). Recently, many authors [25] considered the elliptic systems with subcritical or critical exponents, and they proved the existence of a least energy positive solution or the existence of at least two positive solutions for these problems. In this paper, we construct the k compact Palais-Smale sequences which are suitably localized in correspondence of k maximum points of f. Then we could show that under some assumptions (A1)-(A3), for sufficiently small , there are at least k (≥1) positive solutions of the elliptic system (). By the change of variables
System () is transformed to
Let be the space with the standard norm
Associated with the problem (), we consider the -functional , for ,
Actually, the weak solution of () is the critical point of the functional , that is, satisfies

for any .

We consider the Nehari manifold
(1.1)
where

The Nehari manifold contains all nontrivial weak solutions of ().

Let
(1.2)
then by [[2], Theorem 5], we have
where and is the best Sobolev constant defined by
For the semilinear elliptic systems ()
we define the energy functional , and
If (=1), then we define and

where .

It is well known that this problem
has the unique, radially symmetric and positive ground state solution . Define and , where
Moreover, we have that

This paper is organized as follows. First of all, we study the argument of the Nehari manifold . Next, we prove that the existence of a positive solution of (). Finally, in Section 4, we show that the condition (A2) affects the number of positive solutions of (); that is, there are at least k critical points of such that ((PS)-value) for .

Theorem 1.1 () has at least one positive solution , that is, () admits at least one positive solution.

Theorem 1.2 There exist two positive numbers and such that () has at least k positive solutions for any and , that is, () admits at least k positive solutions.

## 2 Preliminaries

By studying the argument of Han [[7], Lemma 2.1], we obtain the following lemma.

Lemma 2.1 Let (possibly unbounded) be a smooth domain. If , weakly in , and , almost everywhere in Ω, then

Note that is not bounded from below in H. From the following lemma, we have that is bounded from below on .

Lemma 2.2 The energy functional is bounded from below on .

Proof For , by (1.1), we obtain that

where . Hence, we have that is bounded from below on . □

We define
Lemma 2.3 (i) There exist positive numbers σ and such that for ;
1. (ii)

There exists such that and .

Proof (i) By (1.2), the Hölder inequality (, ) and the Sobolev embedding theorem, we have that
where and . Hence, there exist positive σ and such that for .
1. (ii)
For any , since

then . Fix some , there exists such that and . Let . □

Define
Then for , we obtain that
(2.1)
(2.2)

Lemma 2.4 For each , there exists a unique positive number such that and .

Proof Fixed , we consider
Since , , by Lemma 2.3(i), then is achieved at some . Moreover, we have that , that is, . Next, we claim that is a unique positive number such that . Consider
then . Since ,

there exists a unique positive number such that . It follows that . Hence, . □

Remark 2.5 By Lemma 2.3(i) and Lemma 2.4, then for some constant .

Lemma 2.6 Let satisfy

then is a solution of ().

Proof By (2.2), for . Since , by the Lagrange multiplier theorem, there is such that in . Then we have

It follows that and in . Therefore, is a nontrivial solution of () and . □

## 3 (PS) γ -condition in H for

First of all, we define the Palais-Smale (denoted by (PS)) sequence and (PS)-condition in H for some functional J.

Definition 3.1 (i) For , a sequence is a (PS) γ -sequence in H for J if and strongly in as , where is the dual space of H;
1. (ii)

J satisfies the (PS) γ -condition in H if every (PS) γ -sequence in H for J contains a convergent subsequence.

Applying Ekeland’s variational principle and using the same argument as in Cao-Zhou [8] or Tarantello [9], we have the following lemma.

Lemma 3.2 (i) There exists a (PS) -sequence in for .

In order to prove the existence of positive solutions, we want to prove that satisfies the (PS) γ -condition in H for .

Lemma 3.3 satisfies the (PS) γ -condition in H for .

Proof Let be a (PS) γ -sequence in H for such that and in . Then
where , as . It follows that is bounded in H. Hence, there exist a subsequence and such that
Moreover, we have that in . We use the Brézis-Lieb lemma to obtain (3.1) and (3.2) as follows:
(3.1)
(3.2)
Next, we claim that
(3.3)
and
(3.4)
Since , where , then for any , there exists such that . By the Hölder inequality and the Sobolev embedding theorem, we get
Similarly, as . By (A1) and , strongly in , we have that
(3.5)
Let . By (3.1)-(3.4) and Lemma 2.1, we deduce that
and
(3.6)
We may assume that
(3.7)
Recall that
If , by (3.5), then
This implies . By (3.6) and (3.7), we obtain that

which is a contradiction. Hence, , that is, strongly in H. □

## 4 Existence of k solutions

Let be the unique, radially symmetric and positive ground state solution of equation (E 0) in . Recall the facts (or see Bahri-Li [10], Bahri-Lions [11], Gidas-Ni-Nirenberg [12] and Kwong [13]):
1. (i)

for some and ;

2. (ii)
for any , there exist positive numbers , and such that for all

and
By Lien-Tzeng-Wang [14], then
(4.1)
For , we define

Clearly, .

First of all, we want to prove that
Lemma 4.1 For and , then
Moreover, we have that
Proof Part I: Since is continuous in H, , and is uniformly bounded in H for any and , then there exists such that for and any ,
From (A1), we have that . Then
It follows that there exists such that for and any ,
From now on, we only need to show that
Since
and by (4.1), then
(4.2)
For , by (4.2), we have that
Since
then
that is, for and ,
Part II: By Lemma 2.4, there is a number such that , where . Hence, from the result of Part I, we have that for and ,

□

Proof of Theorem 1.1 By Lemma 3.2, there exists a (PS) -sequence in for . Since for and , by Lemma 3.3, there exist a subsequence and such that strongly in H. It is easy to check that is a nontrivial solution of () and . Since and , by Lemma 2.6, we may assume that , . Applying the maximum principle, and in Ω. □

Choosing such that
where and , define and . Suppose for some . Let be given by

where , for and for .

For each , we define

By Lemma 2.4, there exists such that for each . Then we have the following result.

Lemma 4.2 There exists such that if , then for each .

Proof Since
there exists such that

□

We need the following lemmas to prove that for sufficiently small ε, λ, μ.

Lemma 4.3 .

Proof From Part I of Lemma 4.1, we obtain uniformly in i. Similarly to Lemma 2.4, there is a sequence such that and
Let be a minimizing sequence of for . It follows that and

We may assume that and as , where . By the definition of , then . We can deduce that , that is, . □

Lemma 4.4 There exists a number such that if and , then for any .

Proof On the contrary, there exist the sequences and such that , as and for all . It is easy to check that is bounded in H. Suppose that as . Since
then
which is a contradiction. Thus, as . Similarly to the concentration-compactness principle (see Lions [15, 16] or Wang [[6], Lemma 2.16]), then there exist a constant and a sequence such that
(4.3)
where and for some . Let . Then there are a subsequence and such that and weakly in . Using the similar computation of Lemma 2.4, there is a sequence such that and
We deduce that a subsequence satisfies . Then there are a subsequence and such that and weakly in . By (4.3), then and . Applying Ekeland’s variational principle, there exists a (PS) -sequence for and . Similarly to the proof of Lemma 3.3, there exist a subsequence and such that , strongly in and . Now, we want to show that there exists a subsequence such that .
1. (i)
Claim that the sequence is bounded in . On the contrary, assume that , then

1. (ii)
Claim that On the contrary, assume that , that is, . Then use the argument of (i) to obtain that

Since and , strongly in , we have that

Hence, there exists such that if and , then for any . □

Lemma 4.5 If and , then there exists a number such that for any and .

Proof Using the similar computation in Lemma 2.4, we obtain that there is the unique positive number
such that . We want to show that there exists such that if , then for some constant (independent of u and v). First, for ,
Since , then
(4.4)
and
(4.5)
Moreover, we have that
where . It follows that there exists such that for
(4.6)
Hence, by (4.4), (4.5) and (4.6), for some constant (independent of u and v) for . Now, we obtain that
From the above inequality, we deduce that for any and ,
Hence, there exists such that for ,
By Lemma 4.4, we obtain

or for any and . □

Since , then by Lemma 4.3,
(4.7)
By Lemmas 4.1, 4.2 and (4.7), for any () and ,
(4.8)
Applying above Lemma 4.5, we get that
(4.9)
For each , by (4.8) and (4.9), we obtain that
It follows that

Then applying Ekeland’s variational principle and using the standard computation, we have the following lemma.

Lemma 4.6 For each , there is a -sequence in H for .

Proof See Cao-Zhou [8]. □

Proof of Theorem 1.2 For any and , by Lemma 4.6, there is a -sequence for where . By (4.8), we obtain that

Since satisfies the (PS) γ -condition for , then has at least k critical points in for any and . Set and . Replace the terms and of the functional by and , respectively. It follows that () has k nonnegative solutions. Applying the maximum principle, () admits at least k positive solutions. □

## Authors’ Affiliations

(1)
Department of Natural Sciences in the Center for General Education, Chang Gung University

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