Abstract
We consider the perturbed nonlinear elliptic system , , where is the Sobolev critical exponent. Under proper conditions on , , and , the existence result and multiplicity of the system are obtained by using variational method provided is small enough.
1. Introduction and Main Results
In this paper, we are concerned with the existence and multiplicity of nontrivial solutions for the following class of elliptic system: where , denotes the Sobolev critical exponent, is a nonnegative potential, is bounded positive functions, and and are superlinear but subcritical functions.
In the recent years, many papers have considerd the scalar equation which arises in different models; for example, they are related to the existence of standing waves of the nonlinear Schrödinger equation A standing wave of (3) is a solution of the form . We would like to cite the works of Floer and Weinstein [1], Del Pino and Felmer [2, 3], Oh [4], Wang [5], Cingolani and Nolasco [6], Cingolani and Lazzo [7], Ambrosetti et al. [8, 9], Alves and Souto [10, 11], Ding and Lin [12], Liang and Zhang [13], and references therein.
For elliptic systems, we cite the papers of Alves and Soares [14, 15], Alves et al. [16], and Ávila and Yang [17].
Motivated by some results found in [12], a natural question arises whether existence of nontrivial solutions continues to hold for the system (1). We study the existence and multiplicity of the nontrivial solutions for the system (1). Our work completes the results obtained in [12], in the sense that we are working with elliptic systems. To prove our main result, we follow some ideas explored in [12] and also use arguments developed in [13, 18].
In this work, we assume the following assumptions:), , and there is a constant such that the set has finite Lebesgue measure;(), ;() as ;()there exist and such that ()there exist , , such that
Our main results are as follows.
Theorem 1. Assume that and hold. Then, for any , there exists such that ; the perturbed elliptic system (1) has one least energy solution which satisfies
Theorem 2. Let and be satisfied. Moreover, assume that is even in ; then, for any and , there is such that ; the system (1) has at least pairs of solutions which satisfy the estimate (6).
The main difficulty in this paper is the lack of compactness of the energy functional associated to the system (1). To overcome this difficulty, we carefully make estimates and prove that there is a Palais-Smale sequence that has a strongly convergent subsequence. The main results in the present paper can be seen as a complement of studies developed in [12].
This paper is organized as follows. In Section 2, we describe some notations and preliminaries. Section 3 is devoted to the behavior of sequence and the mountain pass level of . Finally, in Section 4, we give the proofs of Theorem 1 and Theorem 2.
2. Preliminaries
Let . The system (1) reads then as We will prove the following result.
Theorem 3. Assume that and hold. Then, for , there exists such that if , the elliptic system (7) has one positive solution of least energy () which satisfies
Theorem 4. Assume that and are satisfied. Moreover, assume that is even in ; then, for any and , there exists such that if , the system (7) has at least pairs of solutions () satisfying the estimate (8).
For the convenience, we quote the necessary notations. The space is a Hilbert space equipped with the inner product and the associated norm . From the assumption , we conclude that embeds continuously in . Moreover, observe that the norm is equivalent to the one for each . It is obvious that, for each , there exists such that if Set and for . The energy functional associated with (7) is defined by where .
Under the assumptions of Theorem 3, standard arguments [18] show that and the critical points of are weak solutions of the elliptic system (7).
3. Technical Lemmas
Lemma 5. If is a sequence for , then and is bounded in .
Proof. From and , we get Together with and , we have that is bounded in and .
Lemma 6. Let . There exists a subsequence such that, for any , there exists with where .
Proof. This proof is similar to the one of Lemma 3.2 [12], so we omit it.
Let satisfying , . if and if . Define , ; then , in .
Lemma 7. One has uniformly in with .
Proof. The proof of Lemma 7 is similar to the one of Lemma 3.3 [12], so we omit it.
Lemma 8. One has along a subsequence
Proof. Since , in and in , we have In connection with the proof of Brezis-Lieb lemma, it is easy to check that Note that and ; we get For any , we have It is standard to check that uniformly in . Together with Lemma 7, we complete the proof of Lemma 8.
Let , ; then in if and only if in . Observe that where . By Lemma 8, we get Meanwhile, by and , there exists such that
Lemma 9. There is a constant independent of such that, for any sequence for with , either or .
Proof. On the contrary, if , we have Let , where is the positive constant in the assumption . In connection with the set which has finite measure and in , we have Therefore, we obtain where is the best Sobolev constant which satisfies By (23), we get Therefore, , where . The proof is completed.
Lemma 10. There is a constant independent of such that if a sequence satisfies then is relatively compact in .
Proof. Lemma 9 implies that satisfies the following local condition. The proof is completed.
Next, we consider and see that has the mountain pass structure.
Lemma 11. Assume that , , and hold. There exist such that
Proof. Observe that . For , there is a constant such that which implies that the required conclusions hold.
Lemma 12. Under the assumptions of Theorem 3, for any finite dimensional subspace , we get
Proof. By and , we get
Since all norms in a finite dimensional space are equivalent and , we easily obtain the desired conclusion.
Define the functional
The standard arguments show that and .
Observe that
For any , there are with and such that , . Set ; then, . For , we get
It is obvious that
In connection with and , , there is such that
for all .
From (39), we easily obtain the following result.
Lemma 13. For any , there exists such that, for each , there exists with ; we get where is defined form Lemma 11.
Proof. This proof is similar to Lemma 4.3 in [12]; it can be easily obtained.
For any , we choose functions such that , , , and . Similarly, one can also get functions with , , , and . Let such that for . Set , , ; then .
Define . For each , we easily get Set and choose some such that Furthermore, we have the following inequality:
Lemma 14. Under the assumptions of Theorem 4, for any and , there is such that, for each , we can get an -dimensional subspace which satisfies
Proof. For any , we choose so small that Meanwhile, we take . By (43), we prove the required conclusion.
4. Proof of the Main Results
Proof of Theorem 3. Define , where . For any , there exists such that, for each , we can choose satisfying .
By Lemma 9, satisfies the condition. Hence, by the mountain pass theorem, there exists satisfying and . So is a weak solution of (7).
Furthermore, we have
This shows that
Proof of Theorem 4. By Lemma 14, for any and , there exists such that, for , we choose an -dimensional subspace with . By Lemma 12, there is such that for all . Define , , where is a version of Benci’s pseudoindex [19].
Since for all (see Lemma 11) and , one has
We easily get that are critical levels and has at least pairs of nontrivial critical points. This proof is completed.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The author would like to thank the anonymous referee for precious comments and suggestions about the original paper. This research was supported by the National Natural Science Foundation of China (11271364).