In this paper, we consider a perturbed p-Laplacian equation with critical nonlinearity and magnetic fields on . By using the variational method, we establish the existence of nontrivial solutions of the least energy.
MSC: 35B33, 35J60, 35J65.
Keywords:p-Laplacian equation; critical nonlinearity; magnetic fields; mountain pass theorem
In this paper, we are concerned with the existence of nontrivial solutions to the following perturbed p-Laplacian equation with critical nonlinearity and magnetic fields of the form
where , i is the imaginary unit, is a real vector potential, , denotes the Sobolev critical exponent and .
This paper is motivated by some works concerning the nonlinear Schr?dinger equation with magnetic fields of the form
where ? is Planck?s constant, i is the imaginary unit, ( ) is the critical exponent, is a real vector potential, and is a scalar electric potential.
In physics, we are interested in the standing wave solutions, that is, solutions to (1.2) of the type
where ? is a sufficiently small constant, E is a real number, and is a complex-valued function satisfying
We can conduct the transition from quantum mechanics to classical mechanics by letting . Thus, the existence of semiclassical solutions has a great charm in physical interest.
Problem (1.3) with has an extensive literature. Different approaches have been taken to investigate this problem under various hypotheses on the potential and nonlinearity. See for example [1-18] and the references therein. The above-mentioned papers mostly concentrated on the nonlinearities with subcritical conditions. Floer and Weinstein in  first studied the existence of single and multiple spike solutions based on the Lyapunov-Schmidt reductions. Subsequently, Oh [16,17] extended the results in a higher dimension. Kang and Wei  established the existence of positive solutions with any prescribed number of spikes, clustering around a given local maximum point of the potential function. In accordance with the Sobolev critical nonlinearities, there have been many papers devoted to studying the existence of solutions to elliptic boundary-valued problems on bounded domains after the pioneering work by Br?zis and Nirenberg . Ding and Lin  first studied the existence of semi-classical solutions to the problem on the whole space with critical nonlinearities and established the existence of positive solutions, as well as of those that change sign exactly once. They also obtained multiplicity of solutions when the nonlinearity is odd.
As far as problem (1.3) in the case of is concerned, we recall Bartsch , Cingolani  and Esteban and Lions . This kind of paper first appeared in . The authors obtained the existence results of standing wave solutions for fixed and special classes of magnetic fields. Cingolani  proved that the magnetic potential only contributes to the phase factor of the solitary solutions for sufficiently small. For more results, we refer the reader to [19-21] and the references therein.
For general , most of the works studied the existence results to equation (1.1) with . See, for example, [22-28] and the references therein. These papers are mostly devoted to the study of the existence of solutions to the problem on bounded domains with the Sobolev subcritical nonlinearities.
However, to our best knowledge, it seems that there is no work on the existence of semiclassical solutions to perturbed p-Laplacian equation on involving critical nonlinearity and magnetic fields. In this paper, we consider problem (1.1) with magnetic fields. The main difficulty in the case is the lack of compactness of the energy functional associated to equation (1.1) because of unbounded domain and critical nonlinearity. At the same time, we must consider complex-valued functions for the appearance of electromagnetic potential . To overcome this difficulty, we chiefly follow the ideas of . Notice that although the ideas were used in other problems, the adaption of the procedure to our problem is not trivial at all. We need to make careful and complex estimates and prove that the energy functional possesses a Palais-Smale sequence, which has a strongly convergent sequence.
We make the following assumptions on , , and throughout the paper:
(V0) , , and there exists such that the set has a finite Lebesgue measure;
(A0) and ;
(K0) , ;
(H1) and uniformly in x as ;
(H2) there are and such that for all ;
(H3) there exist , and such that and for all , where .
Our main result is the following.
Theorem 1Assume that (V0), (A0), (K0) and (H1)-(H3) hold. Then for any , there exists such that if , equation (1.1) has at least one positive least energy solution , which satisfies
The paper is organized as follows. In Section?2, we give some necessary preliminaries. Section?3 is devoted to the technical lemmas. The proof of Theorem?2 is given in the last section.
Let . Equation (1.1) reads then as
We are going to prove the following result.
Theorem 2Assume that (V0), (A0), (K0) and (H1)-(H3) are satisfied. Then for any , there exists such that if , then equation (2.1) has at least one solution of least energy satisfying
In order to prove these theorems, we introduce the space
equipped with the norm
It is known that is the closure of . Similar to the diamagnetic inequality , we have the following inequality
In fact, since is real-valued, one has
(the bar denotes a complex conjugation). This inequality implies that if , then , and, therefore, for any . That is, if in , then in for any and a.e. in .
Solutions of (2.1) will be sought in the Sobolev space as critical points of the functional
It is easy to see that is a -functional on .
3 Behavior of sequence and a mountain pass structure
In this section, we commence by establishing the necessary results which complete the proof of Theorem?2.
Lemma 3.1Let (V0), (A0), (K0) and (H1)-(H3) be satisfied. For the sequence for , we get that and is bounded in the space .
Proof Under assumptions (K0) and (H3), we have
In connection with the facts that and as , we obtain that the sequence is bounded in , and the energy level .??
Next, let denote a sequence. By Lemma?3.1, it is bounded, thus, without loss of generality, we may assume that in . Furthermore, passing to a subsequence, we have in for any and a.e. in .
Lemma 3.2For any , there is a subsequence such that for any , there exists with
Proof It is easily obtained by the similar proof of Lemma?3.2 .??
Let be a smooth function satisfying , if and if . Define . It is not difficult to see that
Lemma 3.3One has
uniformly in with .
Proof By direct computation, we easily obtain in . The local compactness of the Sobolev embedding implies that, for any , we have
uniformly in . For any , there is such that
for all . By the assumptions and the H?lder inequality, we have
This proof is completed.??
Lemma 3.4One has along a subsequence
By the Br?zis-Lieb lemma , we get
We now observe that and , which gives
Moreover, by direct computation, we get
It then follows from the standard arguments that
uniformly in . Combining Lemma?3.3, we get . The proof is completed.??
Let , then . Therefore, in if and only if in .
where . Together with Lemma?3.4, one has
In the following, we consider the energy level of the functional below which the condition holds.
Denote , where b is the positive constant in assumption (V0). Since the set has a finite measure, combining the fact that in , we get
Furthermore, by (K0) and (H1)-(H3), there exists such that
Let S be the best Sobolev constant of the immersion
Lemma 3.5There exists (independent of?) such that, for any sequence for with , either in or .
Proof Arguing by contradiction, assume that , then
Combining the Sobolev inequality, (3.2) and (3.3), we get
which further gives
Denote , then
We obtain the desired conclusion.??
Lemma 3.6There exists a constant (independent of?) such that if a sequence for satisfies , the sequence has a strongly convergent subsequence in .
Proof By the fact that and Lemma?3.5, we easily get the required conclusion.??
Now, we consider and prove that the energy functional possesses the mountain pass structure.
Lemma 3.7Under the assumptions of Theorem?2, there exist such that
Proof The proof of Lemma?3.7 is similar to the one of Lemma?4.1 in .??
Lemma 3.8For any finite dimensional subspace , we have
Proof By assumptions (K0) and (H3), one has
Since all norms in a finite-dimensional space are equivalent, in connection with , we obtain the desired conclusion.??
For ? large enough and small sufficiently, satisfies condition by Lemma?3.6. Furthermore, we will find special finite-dimensional subspace, by which we establish sufficiently small minimax levels.
Define the functional
It is easy to see that and for all . Note that
For any , there is with and such that . Let , then . For any , we have
We derive that
Observe that , and . Therefore, there exists such that for all , we have
Lemma 3.9Under the assumptions of Theorem?2, for any , there is such that for each , there exists with , and
where is defined in Lemma?3.7.
Proof For any , we can choose so small that
Denote and . Let be such that and for all . Then, combining (3.4), meets the requirements.??
4 Proof of Theorem 2
In this section, we give the proof of Theorem?2.
Proof By Lemma?3.9, for any with , we choose and define the minimax value
Lemma?3.6 shows that satisfies condition. Therefore, by the mountain pass theorem, there exists , which satisfies and . That is, is a weak solution of (2.1). Furthermore, it is well known that is the least energy solution of equation (2.1).
Moreover, together with and , we have
By inequality (2.3), we obtain
The proof is complete.??
The authors declare that they have no competing interests.
The authors contributed equally in this article. They 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 was supported by the National Natural Science Foundation of China (11271364) and the Fundamental Research Funds for the Central Universities (2012QNA46).
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