Existence of nontrivial solutions for perturbed p-Laplacian system involving critical nonlinearity and magnetic fields

Under the suitable assumptions, we establish the existence of nontrivial solutions for a perturbed p-Laplacian system in with critical nonlinearity and magnetic fields by using the variational method.

MSC: 35B33, 35J60, 35J65.

In this paper, we consider a class of quasi-linear elliptic systems of the form

where , i is the imaginary unit, is real vector potential, , is a non-negative potential, denotes the Sobolev critical exponent for and is a bounded positive coefficient.

The scalar case corresponding to (1.1) has received considerable attention in recent years. For and , the scalar case corresponding to (1.1) turns into

The equation (1.2) arises in finding standing wave solutions of the nonlinear Schrödinger equation

A standing wave solution of (1.3) is a solution of the form

Then solves (1.3) if and only if solves (1.2) with and .

The equation (1.2) has been extensively investigated in the literature based on various assumptions of the potential and the nonlinearity . See, for example, [1-15] and the references therein.

There are also many works dealing with the magnetic fields and for the scalar case corresponding to (1.1). In [16], the authors firstly obtained the existence of standing waves for special classes of magnetic fields. For many results, we refer the reader to [17-22].

For general , most of the work, as we know, consider the scalar case which corresponds to (1.1) with . See [23-27] and the references therein. We especially mention [24] for the existence of positive solutions for a class of p-Laplacian equations. Gloss [24] studied the existence and asymptotic behavior of positive solutions for quasi-linear elliptic equations of the form

where f is a subcritical nonlinearity without some growth conditions such as the Ambrosetti-Rabinowitz condition. The problem (1.4) has also been studied in [28-32]. The main difficulty in treating this class of equation (1.4) is a possible lack of compactness due to the unboundedness of the domain.

However, to our best knowledge, it seems there is almost no work on the existence of non-trivial solutions to the problem (1.1) involving critical nonlinearity and magnetic fields. We mainly follow the idea of [7]. Observe that though the idea was used in other problems, the adaption of the procedure to the problem is not trivial at all. Because of the appearance of magnetic fields , we must deal with the problem for complex-valued functions and therefore we need more delicate estimates.

The outline of the paper is as follows. The forthcoming section is the main result and preliminary results including the appropriate space setting to work with. In Section 3, we study the behavior of sequence. Section 4 gets that the functional associated to the problem possesses the mountain geometry structure, and the last section concludes the proof of the main result.

Firstly, we make the following assumptions on , , and throughout the paper:

() , and there exists such that the set has finite Lebesgue measure;

() and ;

() , ;

() and as ;

() there exist and such that

() there are , and such that and .

Under the above mentioned conditions, we get the following result.

Theorem 1Suppose that the assumptions (), (), () and ()-() hold. Then for any, there issuch that if, the problem (1.1) has at least one solutionwhich satisfies

Setting , the problem (1.1) is equivalent to the following problem:

We are going to prove the following result.

Theorem 2Suppose that the assumptions (), (), () and ()-() hold. Then for any, there issuch that if, the problem (2.1) has at least one solutionwhich satisfies

For convenience, we quote the following notations. Let denote the Banach space

equipped with the norm

Set and for any .

Similar to the diamagnetic inequality [16], we have

(the bar denotes complex conjugation). This inequality shows that if , then and therefore for any . That is to say, if in , then in for any and a.e. in .

The energy functional associated with (2.1) is defined by

where .

Under the assumptions of Theorem 2, standard arguments [33] show that and its critical points are weak solutions of the equation (2.1).

We call a sequence a sequence if and strongly in ( is the dual space of E). is said to satisfy the condition if any sequence contains a convergent subsequence.

The main result of Section 3 is the following compactness result.

Proposition 3.1Let the assumptions of Theorem 2 be satisfied. There exists a constantindependent ofλsuch that, for anysequenceforwith, eitheror.

As a consequence, we obtain the following result.

Proposition 3.2Assume that the assumptions of Proposition 3.1 hold, satisfies thecondition for all.

In order to prove Proposition 3.1, we need the following lemmas.

Lemma 3.1Let the assumptions of Theorem 2 be satisfied. is asequence of . Thenandis bounded in the spaceE.

Proof

One has

Together with and as , we have

Then is bounded and . □

From Lemma 3.1, we may assume in E and in for any and , a.e. in .

Lemma 3.2Let. There is a subsequencesuch that for any, there iswith

where.

Proof The proof of Lemma 3.2 is similar to that of Lemma 3.2 of [27], so we omit it. □

Let be a smooth function satisfying , if and if . Define , . Obviously, we have

Lemma 3.3One has

and

uniformly inwith.

Proof The local compactness of Sobolev embedding implies that for any , we have

uniformly in . For any , there exists such that

for all . Together with the assumption () and the Hölder inequality, it follows from Lemma 3.2 that

where () are positive constants. Similarly, we can prove

 □

Lemma 3.4Letandbe as defined above. Then the following conclusions hold:

and

Proof By using the similar arguments of [34,35], we have

By (3.1) and the similar idea of proving the Brézis-Lieb lemma [36], it is easy to get

and

Furthermore, using the fact and , we obtain

In order to prove in , for any , it follows that

It is standard to check that

and

uniformly in with . Together with Lemma 3.3, we have

 □

Let , , then , . From (3.1), we get in E if and only if in E.

Observe that

where . Furthermore, we get

Now, we consider the energy level of the functional below which the condition holds.

Let , where b is a positive constant in the assumption (). Since the set has finite measure, we get

In connection with the assumptions ()-() and the Young inequality, there exists such that

Let S be the best Sobolev constant of the immersion

Proof of Proposition 3.1 Assume that , then

and

By the Sobolev embedding inequality and the diamagnetic inequality, we get

This, together with (3.2), gives

Set , then

This completes the proof of Proposition 3.1. □

Proof of Proposition 3.2 Since , we have

In connection with and Proposition 3.1, we complete this proof. □

In the following, we always consider . We will prove that possesses the mountain-pass structure which has been carefully discussed in the works [37,38].

Lemma 4.1Let the assumptions of Theorem 2 be satisfied. There existsuch that

Proof By (3.4), for any , there is such that

Thus,

In connection with , we may choose such that

The fact implies the desired conclusion. □

Lemma 4.2Under the assumptions of Lemma 4.1, for any finite dimensional subspace, we have

Proof Together with the fact , we have

Since all norms in a finite-dimensional space are equivalent and , we complete the proof. □

In the following, we will find special finite-dimensional subspaces by which we establish sufficiently small mini-max levels.

Define the functional

Obviously, it follows that and for all .

Observe that

and

Then, for any , there are with and such that .

Set . Then . For , we get

where

By direct computation, we have

Since , and , we know that there is such that for all , we have

Lemma 4.3For any, there issuch that, there iswith, and

whereis defined in Lemma 4.1.

Proof This proof is similar to that of Lemma 4.3 in [7], so we omit the details. □

Proof By using Lemma 4.3, for any with , we choose and define the mini-max level

where .

By Proposition 3.1, we know that satisfies the condition. Hence, by the mountain-pass theorem, there is such that and . This shows is a weak solution of (2.1).

Moreover, note that and . Then

Furthermore, together with the diamagnetic inequality, we prove that satisfies the estimate (2.2). The proof is complete. □

The authors declare that they have no competing interests.

The authors contributed equally in this article. All authors read and approved the final manuscript.

The authors would like to appreciate the referees for their precious comments and suggestions about the original manuscript. This research is supported by the National Natural Science Foundation of China (11271364) and the Fundamental Research Funds for the Central Universities (2012QNA46).

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