Abstract
Keywords:
pLaplacian system; critical nonlinearity; magnetic fields; variational method1 Introduction
In this paper, we consider a class of quasilinear elliptic systems of the form
where , i is the imaginary unit, is real vector potential, , is a nonnegative 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, [115] 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 [1722].
For general , most of the work, as we know, consider the scalar case which corresponds to (1.1) with . See [2327] and the references therein. We especially mention [24] for the existence of positive solutions for a class of pLaplacian equations. Gloss [24] studied the existence and asymptotic behavior of positive solutions for quasilinear elliptic equations of the form
where f is a subcritical nonlinearity without some growth conditions such as the AmbrosettiRabinowitz condition. The problem (1.4) has also been studied in [2832]. 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 nontrivial 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 complexvalued 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.
2 Main results and preliminaries
Firstly, we make the following assumptions on , , and throughout the paper:
() , and there exists such that the set has finite Lebesgue measure;
() 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
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
Under the assumptions of Theorem 2, standard arguments [33] show that and its critical points are weak solutions of the equation (2.1).
3 condition
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
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
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
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 BrézisLieb 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
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
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. □
4 The mountainpass structure
In the following, we always consider . We will prove that possesses the mountainpass 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 finitedimensional space are equivalent and , we complete the proof. □
In the following, we will find special finitedimensional subspaces by which we establish sufficiently small minimax levels.
Define the functional
Obviously, it follows that and for all .
Observe that
and
Then, for any , there are with and such that .
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
Proof This proof is similar to that of Lemma 4.3 in [7], so we omit the details. □
5 Proof of Theorem 2
Proof By using Lemma 4.3, for any with , we choose and define the minimax level
By Proposition 3.1, we know that satisfies the condition. Hence, by the mountainpass 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. □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors contributed equally in this article. All authors read and approved the final manuscript.
Acknowledgements
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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