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Multiplicity result for a critical elliptic system with concave-convex nonlinearities
Boundary Value Problems volume 2013, Article number: 268 (2013)
Abstract
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.
1 Introduction
In this paper, we study the existence and multiplicity of solutions for the following elliptic system of noncooperative type:
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]).
It is well known that weak solutions of () are critical points of the following functional defined on the Hilbert space :
We call a weak solution of () if and
for every .
The celebrated papers by Brezis and Nirenberg [3] and by Ambrosetti et al. [4] have inspired research on differential equations and systems with critical terms, and with concave-convex nonlinearities, respectively, see, for example, [5–12] and the many references therein. In recent years, more and more attention have been paid to the existence and multiplicity of solutions of the following elliptic system of cooperative type:
In [9], 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 [13], 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 [15] 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 [18], 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.
We fix an orthonormal basis of Y, and we consider on the τ-topology introduced by Kryszewski and Szulkin in [19]; that is, the topology associated to the following norm
Clearly . Moreover, if is a bounded sequence in X, then
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)
Let be an even functional which is τ-upper semicontinuous, and such that ∇φ is τ-weak sequentially compact in for any . Let with , and , such that
Then there exists such that
-
(i)
if or if ,
-
(ii)
,
-
(iii)
, ,
-
(iv)
is non increasing, ,
-
(v)
Each point has a τ-neighborhood such that is contained in a finite-dimensional subspace of X,
-
(vi)
η is τ-continuous,
-
(vii)
is odd ,
where and .
Proof We define
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 [18] with . □
We introduce the following notations:
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
(B1) ,
(B2) ,
(B3) , ,
(B4) Φ satisfies the condition .
Then Φ has a sequence of critical points such that and as .
Proof Let be the set of maps such that
-
(a)
γ is odd, τ-continuous and ,
-
(b)
each has a τ-neighborhood in such that is contained in a finite dimensional subspace of X,
-
(c)
.
Define
It follows from the definitions that . Lemma 10 in [18] implies that , which in turn implies that .
Let , , and let such that
We claim that
Hence, there exists a sequence such 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.
We proceed by contradiction by assuming that (3) does not hold, and then, we apply Lemma 4 with , and . We may assume that . Next, by following the proof of Theorem 11 in [18], one can easily verify that the map β defined on by belongs to . Now, (2) and (ii) of Lemma 4 imply that
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 .
The functional J defined by (1) then reads as follows:
A standard argument shows that J is of class on X and
Lemma 7 Let such that in X. Then there exists a subsequence such that .
Proof Let . Using (5), we have
It is clear that
Now up to a subsequence we have
By the Sobolev embedding theorem and are bounded in , and . So (resp. ) is bounded in (resp. in ), and are bounded in .
By Theorem 10.36 in [20], we have
Since , it follows that
and for every . Hence, . □
Lemma 8 The functional J satisfies the Palais-Smale condition.
Proof Let such that
We deduce from (5) that
Hence for n big enough, we have
which implies, since , that is bounded.
On the other hand, by using (4) and (5), we obtain
Since is bounded, and
for n big enough, we easily deduce that is bounded.
Consequently, we have, up to a subsequence,
On the other hand, and equation (6) imply that and that
Now, because of equation (7), it follows that
Let us combine the preceding equations in order to cancel the integrals. We then get
But since equation (6) leads to
it is then easy to conclude that
Hence . □
Now, we select an orthonormal basis of , and we consider the τ-topology on , where Y and Z are defined by
Lemma 9 J is τ-lower semicontinuous, and ∇J is τ-weak sequentially compact in for any .
Proof
-
1.
Let and such that in X and . By the definition of τ, converges strongly to v in . Clearly, we have
which 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
that .
-
2.
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 .
□
Proof of Theorem 1 We recall that
Let . Then, since , we have
where
Therefore, for every such that , we have
On the other hand, it is clear that
Hence, for every such that , we have
We know by [21] that as , hence we deduce that as .
We have then proved that assumptions (B1) and (B3) of Theorem 6 are satisfied.
Let . Since the norms and are equivalent on , and continuously embeds into , we have
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. □
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Acknowledgements
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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Batkam, C.J., Colin, F. Multiplicity result for a critical elliptic system with concave-convex nonlinearities. Bound Value Probl 2013, 268 (2013). https://doi.org/10.1186/1687-2770-2013-268
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DOI: https://doi.org/10.1186/1687-2770-2013-268