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
For , , and , we consider the semilinear elliptic systems
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
(A3) where , and .
In , 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 [2-5] 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
The Nehari manifold contains all nontrivial weak solutions of ( ).
then by [, 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
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 1.2There exist two positive numbers and such that ( ) has at leastkpositive solutions for any and , that is, ( ) admits at leastkpositive solutions.
By studying the argument of Han [, Lemma 2.1], we obtain the following lemma.
Lemma 2.1Let (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 2.2The 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 . □
Lemma 2.3 (i) There exist positive numbersσand such that for ;
(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 .
(ii) For any , since
then . Fix some , there exists such that and . Let . □
Then for , we obtain that
Lemma 2.4For 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 2.6Let 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;
(ii) J satisfies the (PS)γ-condition in H if every (PS)γ-sequence in H for J contains a convergent subsequence.
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 inHfor .
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:
Next, we claim that
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
Let . By (3.1)-(3.4) and Lemma 2.1, we deduce that
We may assume 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
(i) for some and ;
(ii) for any , there exist positive numbers , and such that for all
By Lien-Tzeng-Wang , then
For , we define
First of all, we want to prove that
Lemma 4.1For 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
and by (4.1), then
For , by (4.2), we have that
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 4.2There exists such that if , then for each .
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 4.4There 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
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 .
(i) Claim that the sequence is bounded in . On the contrary, assume that , then
which is a contradiction.
(ii) Claim that On the contrary, assume that , that is, . Then use the argument of (i) to obtain that
which is a contradiction.
Since and , strongly in , we have that
which is a contradiction.
Hence, there exists such that if and , then for any . □
Lemma 4.5If 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
Moreover, we have that
where . It follows that there exists such that for
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,
By Lemmas 4.1, 4.2 and (4.7), for any ( ) and ,
Applying above Lemma 4.5, we get that
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 4.6For each , there is a -sequence inHfor .
Proof See Cao-Zhou . □
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. □
The author declares that he has no competing interests.
The author was grateful for the referee’s helpful suggestions and comments.
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