Abstract
Keywords:
pLaplacian equation; critical nonlinearity; magnetic fields; mountain pass theorem1 Introduction
In this paper, we are concerned with the existence of nontrivial solutions to the following perturbed pLaplacian 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 complexvalued 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 [118] and the references therein. The abovementioned papers mostly concentrated on the nonlinearities with subcritical conditions. Floer and Weinstein in [11] first studied the existence of single and multiple spike solutions based on the LyapunovSchmidt reductions. Subsequently, Oh [16,17] extended the results in a higher dimension. Kang and Wei [14] 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 boundaryvalued problems on bounded domains after the pioneering work by Br?zis and Nirenberg [4]. Ding and Lin [8] first studied the existence of semiclassical 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 [2], Cingolani [5] and Esteban and Lions [10]. This kind of paper first appeared in [10]. The authors obtained the existence results of standing wave solutions for fixed and special classes of magnetic fields. Cingolani [5] 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 [1921] and the references therein.
For general , most of the works studied the existence results to equation (1.1) with . See, for example, [2228] 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 pLaplacian 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 complexvalued functions for the appearance of electromagnetic potential . To overcome this difficulty, we chiefly follow the ideas of [5]. 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 PalaisSmale sequence, which has a strongly convergent sequence.
We make the following assumptions on , , and throughout the paper:
(V_{0}) , , and there exists such that the set has a finite Lebesgue measure;
(H_{1}) and uniformly in x as ;
(H_{2}) there are and such that for all ;
(H_{3}) there exist , and such that and for all , where .
Our main result is the following.
Theorem 1Assume that (V_{0}), (A_{0}), (K_{0}) and (H_{1})(H_{3}) hold. Then for any, there existssuch 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.
2 Preliminaries
Let . Equation (1.1) reads then as
We are going to prove the following result.
Theorem 2Assume that (V_{0}), (A_{0}), (K_{0}) and (H_{1})(H_{3}) are satisfied. Then for any, there existssuch that if, then equation (2.1) has at least one solution of least energysatisfying
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 [10], we have the following inequality
In fact, since is realvalued, 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 [29].
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 (V_{0}), (A_{0}), (K_{0}) and (H_{1})(H_{3}) be satisfied. For thesequencefor, we get thatandis bounded in the space.
Proof Under assumptions (K_{0}) and (H_{3}), 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 subsequencesuch that for any, there existswith
Proof It is easily obtained by the similar proof of Lemma?3.2 [8].??
Let be a smooth function satisfying , if and if . Define . It is not difficult to see that
Lemma 3.3One has
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
and
Proof Combining Lemma?2.1 of [30] and the arguments of [31], one has
By the Br?zisLieb lemma [32], we get
and
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 .
Note that
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 (V_{0}). Since the set has a finite measure, combining the fact that in , we get
Furthermore, by (K_{0}) and (H_{1})(H_{3}), there exists such that
Let S be the best Sobolev constant of the immersion
Lemma 3.5There exists (independent of?) such that, for anysequenceforwith, eitherinor.
Proof Arguing by contradiction, assume that , then
Combining the Sobolev inequality, (3.2) and (3.3), we get
which further gives
We obtain the desired conclusion.??
Lemma 3.6There exists a constant (independent of?) such that if asequenceforsatisfies, the sequencehas 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 existsuch that
Proof The proof of Lemma?3.7 is similar to the one of Lemma?4.1 in [8].??
Lemma 3.8For any finite dimensional subspace, we have
Proof By assumptions (K_{0}) and (H_{3}), one has
Since all norms in a finitedimensional 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 finitedimensional 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
where
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 issuch that for each, there existswith, and
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.??
Competing interests
The authors declare that they have no competing interests.
Authors? contributions
The authors contributed equally in this article. They 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 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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