Multiple positive solutions for semilinear elliptic systems involving subcritical nonlinearities in
© Lin; licensee Springer 2012
Received: 29 March 2012
Accepted: 4 October 2012
Published: 24 October 2012
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.
Let f, g and h satisfy the following conditions:
(A1) f is a positive continuous function in and .
(A3) where , and .
for any .
The Nehari manifold contains all nontrivial weak solutions of ().
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.
By studying the argument of Han [, Lemma 2.1], we obtain the following lemma.
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 .
where . Hence, we have that is bounded from below on . □
There exists such that and .
- (ii)For any , since
then . Fix some , there exists such that and . Let . □
Lemma 2.4 For each , there exists a unique positive number such that and .
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 .
then is a solution of ().
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.
J satisfies the (PS) γ -condition in H if every (PS) γ -sequence in H for J contains a convergent subsequence.
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 .
which is a contradiction. Hence, , that is, strongly in H. □
4 Existence of k solutions
for some and ;
- (ii)for any , there exist positive numbers , and such that for all
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. □
where , for and for .
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 .
We need the following lemmas to prove that for sufficiently small ε, λ, μ.
Lemma 4.3 .
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 .
- (i)Claim that the sequence is bounded in . On the contrary, assume that , then
- (ii)Claim that On the contrary, assume that , that is, . Then use the argument of (i) to obtain that
which is a contradiction.
which is a contradiction.
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 .
or for any and . □
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 . □
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. □
The author was grateful for the referee’s helpful suggestions and comments.
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