Existence of nontrivial solutions for perturbed p-Laplacian system involving critical nonlinearity and magnetic fields
© Zhang et al.; licensee Springer. 2013
Received: 5 October 2012
Accepted: 9 January 2013
Published: 24 January 2013
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.
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.
Then solves (1.3) if and only if solves (1.2) with and .
There are also many works dealing with the magnetic fields and for the scalar case corresponding to (1.1). In , 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].
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 . 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.
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;
() and ;
() , ;
() and as ;
() there are , and such that and .
Under the above mentioned conditions, we get the following result.
We are going to prove the following result.
Set and for any .
(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 .
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 3.1 Let the assumptions of Theorem 2 be satisfied. There exists a constant independent of λ such that, for any sequence for with , either or .
As a consequence, we obtain the following result.
Proposition 3.2 Assume that the assumptions of Proposition 3.1 hold, satisfies the condition for all .
In order to prove Proposition 3.1, we need the following lemmas.
Lemma 3.1 Let the assumptions of Theorem 2 be satisfied. is a sequence of . Then and is bounded in the space E.
Then is bounded and . □
From Lemma 3.1, we may assume in E and in for any and , a.e. in .
Proof The proof of Lemma 3.2 is similar to that of Lemma 3.2 of , so we omit it. □
uniformly in with .
Let , , then , . From (3.1), we get in E if and only if in E.
Now, we consider the energy level of the functional below which the condition holds.
This completes the proof of Proposition 3.1. □
In connection with and Proposition 3.1, we complete this proof. □
4 The mountain-pass structure
The fact implies the desired conclusion. □
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.
Obviously, it follows that and for all .
Then, for any , there are with and such that .
where is defined in Lemma 4.1.
Proof This proof is similar to that of Lemma 4.3 in , so we omit the details. □
5 Proof of Theorem 2
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).
Furthermore, together with the diamagnetic inequality, we prove that satisfies the estimate (2.2). The proof is complete. □
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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