- Open Access
Multiplicity result for a critical elliptic system with concave-convex nonlinearities
© Batkam and Colin; licensee Springer. 2013
Received: 10 April 2013
Accepted: 29 August 2013
Published: 6 December 2013
We study the existence of multiple solutions of a strongly indefinite elliptic system involving the critical Sobolev exponent and concave-convex nonlinearities. By using a suitable version of the dual fountain theorem established in this paper, we prove the existence of infinitely many small energy solutions.
MSC:35A02, 35J50, 35J60.
where Ω is a bounded smooth domain in with , , and , and satisfy , where denotes the critical Sobolev exponent. The solutions of () are steady states of reaction-diffusion systems, which serve as a class of models with applications in physics, chemistry and biology (see, for instance, [1, 2]).
for every .
In , by extracting Palais-Smale sequences on the Nehari manifold, Hsu and Lin proved that if λ and μ are small, then () has at least two solutions. Subsequently, their result was extended by Chen and Wu , who considered () with sign-changing weight functions. See also [12, 14] and the references therein for some related results.
However, what can be said about the noncooperative case? To the best of our knowledge, there is no result on problem (), even in the subcritical case . In addition to the combined effects of the concave and convex terms, two main difficulties arise when studying (). The first one is that the energy functional J above is strongly indefinite in the sense that it is neither bounded from above nor from below, even on subspaces of finite codimension. Therefore, the usual critical point theorems such as the mountain pass theorem cannot be used. Moreover, the method of the Nehari manifold, which is extensively used in the cooperative case does not apply in this situation. We refer to  for a unified approach on the method of the Nehari manifold. The second difficulty arises with the critical Sobolev exponent. Indeed, it is well known that the embedding is not compact because of the action of dilatations. Therefore, the energy functional is not expected to satisfy the Palais-Smale condition (see Definition 5 below ). Usually, the best we can expect is to find a bounded from above subset such that the functional satisfies the Palais-Smale condition at every level , see, for instance, [3, 9, 16, 17]. But surprisingly enough, we will show that in our case the functional J satisfies the Palais-Smale condition.
The main result of the paper is the following.
Theorem 1 Let . Then problem () has a sequence of solutions such that and as .
Remark 2 Theorem 1 remains true in the subcritical case, i.e., if we assume that if or if . In this case, the proof of Theorem 1 is considerably simplified, since the embedding is now compact.
The paper is organized as follows. In order to prove Theorem 1, we need a new critical point theorem for strongly indefinite functionals, which will be provided in Section 2. This critical point theorem generalizes the dual fountain theorem of Bartsch and Willem, and is not based on any reduction method. It should be noted that problem () does not fit into the framework of the theorem proved in , which is only suitable for finding large energy solutions. Finally the proof of our main result is presented in Section 3.
2 Critical point theory
Let Y be a closed subspace of a separable Hilbert space X endowed with the inner product and the associated norm . We denote by and the orthogonal projections.
Definition 3 Let and , . We say that is τ-weak sequentially compact in if for every sequence which τ-converges to u in , there is a subsequence such that .
Lemma 4 (Deformation lemma)
if or if ,
is non increasing, ,
Each point has a τ-neighborhood such that is contained in a finite-dimensional subspace of X,
η is τ-continuous,
is odd ,
where and .
We claim that for every , there exists a τ-open neighborhood of v such that ( can be chosen to be symmetric, that is, ). In fact, if this is not true, then we can find a sequence which τ-converges to an element and such that . Since ∇φ is τ-weak sequentially compact, there is a subsequence such that , and this implies that , which is in contradiction with the definition of w.
The rest of the proof follows the same lines as the proof of Lemma 8 in  with . □
Before we state and prove our critical point theorem, we recall the following definition.
Definition 5 A functional is said to satisfy the Palais-Smale condition (resp. condition ) if every sequence such that is bounded (resp. ) and , has a convergent subsequence.
Theorem 6 (Generalized dual fountain theorem)
Let be an even functional which is τ-lower semicontinuous and such that ∇Φ is τ-weak sequentially compact in for any . If, for every , there exist such that
(B3) , ,
(B4) Φ satisfies the condition .
Then Φ has a sequence of critical points such that and as .
γ is odd, τ-continuous and ,
each has a τ-neighborhood in such that is contained in a finite dimensional subspace of X,
It follows from the definitions that . Lemma 10 in  implies that , which in turn implies that .
The conclusion of Theorem 6 then follows from (B4) and (B3).
To complete the proof of Theorem 6, it remains to show that (3) holds.
which contradicts the definition of . □
3 Proof of the main results
In this section, we denote by the usual norm.
We consider the Sobolev space endowed with the norm and with the product norm .
Lemma 7 Let such that in X. Then there exists a subsequence such that .
By the Sobolev embedding theorem and are bounded in , and . So (resp. ) is bounded in (resp. in ), and are bounded in .
and for every . Hence, . □
Lemma 8 The functional J satisfies the Palais-Smale condition.
which implies, since , that is bounded.
for n big enough, we easily deduce that is bounded.
Hence . □
Lemma 9 J is τ-lower semicontinuous, and ∇J is τ-weak sequentially compact in for any .
- 1.Let and such that in X and . By the definition of τ, converges strongly to v in . Clearly, we havewhich implies, since is bounded and , that is bounded. Up to a subsequence, we may assume that in and in , and for almost every . By using Fatou’s lemma and the weak lower semicontinuity of the norm , we deduce from the inequality
Assume that in , . The same argument as above shows that is bounded, and then . It follows from Lemma 7 that ∇J is τ-weak sequentially compact in .
We know by  that as , hence we deduce that as .
We have then proved that assumptions (B1) and (B3) of Theorem 6 are satisfied.
where and are constants. It is then easy to verify, since , that assumption (B2) of Theorem 6 is satisfied for small enough.
By using Lemmas 8, 9, we can apply Theorem 6 and get the desired result. □
The authors are grateful to the anonymous referees for their careful reading of the paper and for helpful comments. This work was funded by a grant from the Natural Sciences and Engineering Research Council of Canada.
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