Abstract

We consider a perturbed quasilinear elliptic system involving the -Laplacian with critical growth terms in . Under proper conditions, we establish the existence of nontrivial solutions by using the variational methods.

1. Introduction

In this paper, we are concerned with the following perturbed quasilinear elliptic system involving the -Laplacian: where , is the -Laplacian operator, , satisfy , is the critical Sobolev exponent for , and , , , satisfy the following assumptions:(), , and there exists such that the set has finite Lebesgue measure;();() and are bounded and positive functions.

Set , , and . The problem (1) reduces to the semilinear scalar quasilinear elliptic equation The type of problem (2) including and has been extensively studied in many papers involving bounded domain and unbounded domain. See, for example, [1–13] and the references therein.

In recent years, much attention has been paid to the existence of solutions for problem (1) with and in bounded domain. Wu [14] was concerned with the following semilinear elliptic system with subcritical nonlinearity of concave-convex type and sign-changing weights: where , , satisfy and the functions , , satisfy some suitable conditions. He established the existence of at least two positive solutions for the problem (3) when the pair of the parameters belongs to a certain subset of . Hsu and Lin [15] considered a similar problem and proved that the problem (3) has at least two positive solutions in involving critical exponents. Subsequently, Hsu [16] extended the results of [15] to the quasilinear case . The paper [16] was devoted to the following quasilinear elliptic system: where , , satisfy and denotes the critical Sobolev exponent. He proved that the problem (4) has at least two positive solutions in .

However, as far as we know, there are almost no results on problem (1) involving critical exponents in whole space. In our work, we consider the problem (1) and use variational methods to get positive solutions. Our main arguments use similar ideas found in [8, 15]. The main difficulty is that some estimates and results hold for the Laplacian operator but not for -Laplacian operator. At the same time, the corresponding functional to problem (1) lacks compactness because of unbounded domain and critical exponent. We can prove the functional associated to (1) possesses condition at some energy level . The main result in the paper can be seen as a complement of results obtained in [8, 16].

In particular, we will mention our own work [17] and furthermore discuss the differences between these two papers. In [17], we are concerned with the following system: The coupled system (5) shows that the coupled terms are and . The energy functional associated with (5) is defined by We can prove that satisfies the condition at some energy level and possesses the mountain-pass structure. By using the mountain-pass theorem, we obtain the existence of nontrivial weak solutions for the system (5). In [17], we mostly focus on discussing the properties of the functions , and the associated primitive function which bring some difficulties in proving that satisfies the compactness condition. The difficulty is not mainly due to the critical nonlinearities and .

But, in the current paper, the coupled terms of the system (1) are the critical nonlinearities and (). By using the variational methods, we can establish the existence of nontrivial weak solutions. Although we use similar ideas found in [17], the difficulty is mostly due to the effect of the coupled critical nonlinearities. By means of best Sobolev embedding constant and Holder inequality, we find some energy level and prove that the corresponding energy functional associated with the system (1) satisfies the condition for all . Comparing with the procedure in [17], the one in this paper is complex.

Let . Problem (1) is equivalent to the following problem: We will prove that problem (7) has at least one nontrivial solution under the suitable conditions on , , , .

Set the space and the associated norm for any . Let . Thus . The problem (7) is posed in the framework of the Sobolev space with the norm We will show the existence of nontrivial solutions of (7) by searching for critical points of the associated functional: In fact, the weak solutions of (7) are the critical points of the functional . As to the weak solution of (7), we mean that which satisfies The main result of this paper reads as follows.

Theorem 1. Let ()–() be satisfied. Then, for any , there is such that if , the problem (7) has at least one solution which satisfies This paper is organized as follows. In Section 2, we give some notations and preliminaries. Section 3 is devoted to the proof of the main result.

2. Notations and Preliminaries

In this section, we will show the range of where the condition holds for the functional and prove that possesses the mountain-pass structure. First, we make use of the following notations.

Let denote the collection of smooth functions with compact support.

Let be the completion of under the norm , , denote Lebesgue spaces and the norm of is denoted by for . The dual space of a Banach space will be denoted by . is the ball in . denotes as . represent various positive constants, the exact values of which are not important.

is the best Sobolev embedding constant defined by By use of a similar proof of Theorem 5 in [18], we can obtain that where is the best Sobolev embedding constant defined by which is achieved by the function

Definition 2. Let .(1)A sequence which satisfies and strongly in as is called a sequence in for .(2)We say that satisfies condition if and only if any sequence in for has a convergent subsequence.
The main result of Section 2 is the following compactness result.

Proposition 3. Assume that ()–() are satisfied. Then, for any sequence for , there exists a constant (independent of ) such that either or ; that is to say, the functional satisfies the condition for all .

To prove Proposition 3, we need the following lemmas.

Lemma 4. Assume that ()–() hold. Let the sequence be a sequence for ; then we get that and is bounded in the space .

Proof. By direct computation, we have Together with () and , we get By the fact that and , we easily obtain the desired conclusion.

Lemma 4 shows that a sequence is bounded. Hence, we may assume that in , , a.e. in and in for any .

Lemma 5. We can choose a subsequence such that, for any , there is with : where .

Proof. Note that, for each , we have So there exists such that for all . We may choose such that It is obvious that there is satisfying Furthermore the lemma follows.

Let be a smooth function satisfying if and if . Define and . It is obvious that

Lemma 6. One has uniformly in with .

Proof. Because the proof is similar to the one of Lemma 3.4 [8], we omit it.

Lemma 7. One has along a subsequence

Proof. Together with the fact that in , we get By using similar ideas of proving the Brézis-Lieb Lemma [19], we easily get In connection with the fact that and , we obtain For any , it follows that It is standard to check uniformly in .
Combining Lemma 6 and , we complete the proof of Lemma 7.

Set and ; then and . In order to check in , we only prove in .

Observe that where . By Lemma 7, we get In addition, by where . Furthermore, by using Holder inequality, , , and , for any , there is a constant such that Thus Together with (38), we have Set . This implies . In the following, we give the proof of Proposition 3.

Proof of Proposition 3. For any sequence with , it follows that either or . On the contrary, if , this shows In connection with the above-mentioned analysis, we get that the functional satisfies the condition for all .

3. Proof of the Main Result

First, we will prove that the functional possesses the mountain-pass structure.

Lemma 8. Assume that ()–() be satisfied. There exist , such that

Proof. Observe that, for each , there is such that if , Together with Young inequality, we get Combining () and (46), there is a constant such that Set ; it implies We complete the proof.

Lemma 9. Under the assumptions of Lemma 8, for any finite-dimensional subspace , we have

Proof. Together with the assumptions () and (), it follows that where .
Noting that all norms in a finite-dimensional space are equivalent and , Lemma 9 follows.

Next, we will search for special finite-dimensional subspaces by which we establish sufficiently small minimax levels.

Define the functionals It is apparent that and for all .

Observe that For any , there are with such that Let ; then . For , we get Combining and , this implies that there is such that, for all , we have Thus, for , It follows from (56) that we have the following.

Lemma 10. For any , there is such that, for each , there exists with , and where is defined in Lemma 8.

Proof. For any , we can choose so small that Set . Taking , there is such that and for all . By (56), we choose which satisfies the requirements.

Finally, we will give the proof of the main result.

Proof of Theorem 1. Denote by where .
By Lemma 10, for any with , there is such that, for , we choose satisfying .
It is clear that the functional satisfies condition and has the mountain-pass structure if . Hence, by the mountain-pass theorem, there is such that Namely, is a weak solution of (7). Similar to the arguments in [8], we also get that is a positive least energy solution. Furthermore, This shows that The proof is complete.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

The authors contributed equally to this paper. They read and approved the final paper.

Acknowledgments

The authors would like to thank the referees for their precious comments and suggestions about the original paper. This research was supported by the Fundamental Research Funds for the Central Universities (2013XK03).