Existence of Nontrivial Solutions for Perturbed Elliptic System in R N

and Applied Analysis 3 Together with J λ (u n , V n ) → c and J λ (u n , V n ) → 0, we have that {(u n , V n )} is bounded in E and c ≥ 0. Lemma 6. Let d ∈ [2, 2). There exists a subsequence {(u n i , V n i )} such that, for any ε > 0, there exists r ε > 0 with


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  ≥ 3, 2 * = 2/( − 2) denotes the Sobolev critical exponent, () is a nonnegative potential, () is bounded positive functions, and   (, V) and  V (, V) are superlinear but subcritical functions.
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  > 0 such that the set ]  = { ∈ R  : () < } has finite Lebesgue measure; Our main results are as follows.
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 (PS) sequence and the mountain pass level of   .Finally, in Section 4, we give the proofs of Theorem 1 and Theorem 2.

Preliminaries
Let  =  −2 .The system (1) reads then as We will prove the following result.
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
In connection with the proof of Brezis-Lieb lemma, it is easy to check that lim Note that   (  , V  ) →  and   (ũ  , Ṽ ) →   (, V); we get For any (, ) ∈ , we have It is standard to check that lim uniformly in ‖(, )‖  ≤ 1. Together with Lemma 7, we complete the proof of Lemma 8.
where  min = inf ∈R  () > 0. By Lemma 8, we get Meanwhile, by ( 2 ) and ( 3 ), there exists Lemma 9.There is a constant  0 > 0 (independent of ) such that, for any ( Proof.On the contrary, if (  , V  )   (, V), we have lim inf Let   () = max{(), }, where  is the positive constant in the assumption ( 0 ).In connection with the set ]  which has finite measure and Therefore, we obtain where  is the best Sobolev constant which satisfies By ( 23), we get Therefore,  0  1−/2 ≤  −   (, V) +  (1), where Proof.Lemma 9 implies that   satisfies the following local (PS)  condition.The proof is completed.
Lemma 12.Under the assumptions of Theorem 3, for any finite dimensional subspace  ⊂ , we get Proof.By ( 0 ) and ( 3 ), we get Since all norms in a finite dimensional space are equivalent and ,  > 2, we easily obtain the desired conclusion.
Define the functional The standard arguments show that Φ  ∈  .
Lemma 13.For any  > 0, there exists Λ  > 0 such that, for each  ≥ Λ  , 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 Set   := max{‖(   ,    )‖ Furthermore, we have the following inequality: max Lemma 14.Under the assumptions of Theorem 4, for any  ∈ N and  > 0, there is Λ  > 0 such that, for each  ≥ Λ  , we can get an -dimensional subspace  which satisfies max Proof.For any  > 0, we choose  > 0 so small that Meanwhile, we take  =    .By (43), we prove the required conclusion.
Furthermore  We easily get that    are critical levels and   has at least  pairs of nontrivial critical points.This proof is completed.