Multiplicity of solutions of perturbed Schrödinger equation with electromagnetic fields and critical nonlinearity in
© Liang and Song; licensee Springer. 2014
Received: 11 June 2014
Accepted: 3 November 2014
Published: 27 November 2014
In this paper, we deal with the existence and multiplicity of solutions for perturbed Schrödinger equation with electromagnetic fields and critical nonlinearity in : for all , where , is a nonnegative potential. By using Lions’ second concentration compactness principle and concentration compactness principle at infinity to prove that the condition holds locally and by variational method, we show that this equation has at least one solution provided that , for any , it has m pairs of solutions if , where ℰ and are sufficiently small positive numbers.
MSC: 35J60, 35B33.
where . Here, i is the imaginary unit, denotes the Sobolev critical exponent and , and are functions satisfying some conditions.
where ħ is Plank constant, is a real vector (magnetic) potential with magnetic field and is a scalar electric potential.
where and . The transition from quantum mechanics to classical mechanics can be formally performed by letting . Thus the existence of solutions for ħ small, semi-classical solutions, has important physical interest.
It is well known that the linear Schrödinger equation is a basic tool of quantum mechanics, and it provides a description of the dynamics of a particle in a non-relativistic setting. The nonlinear Schrödinger equation arises in different physical theories, e.g., the description of Bose-Einstein condensates and nonlinear optics, see  and the references cited there. Both the linear and the nonlinear Schrödinger equations have been widely considered in the literature. The main purpose of this paper is to study the existence and multiplicity of solutions of perturbed Schrödinger equations with electromagnetic fields and critical nonlinearity (1.1).
Problem (1.3) with has an extensive literature. Different approaches have been taken to attack this problem under various hypotheses on the potential and the nonlinearity. See, for example, – and the references therein. Observe that in all these papers the nonlinearities are assumed to be subcritical. In , using a Lyapunov-Schmidt reduction, Floer and Weinstein established the existence of single and multiple spike solutions. Their method and results were later generalized by Oh  to the higher-dimensional case. 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 Brezis 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.
By using the linking theorem twice to the corresponding functional, they established the existence results. Chabrowski and Szulkin  considered problem (1.4) under assumption that changes sign; by using a min-max type argument based on a topological linking, they obtained a solution in the Sobolev space which was defined in the paper. Assume , Han  studied problem (1.4) and established the existence of nontrivial solutions in the critical case by means of variational method. For more results, we refer the reader to , – and the references therein.
In the present paper, we consider the existence of solutions for problem (1.1) under the condition and critical nonlinearity. It seems that Byeon and Wang  were the first to study energy level and the asymptotic behavior of positive solutions to Schrödinger equations under the condition . In , Cao and Tang extended the results of Byeon and Wang . However, to the best knowledge, it seems that there are few works on the existence of solutions to be the problems on involving critical nonlinearities with electromagnetic fields. We mainly follow the idea of , . Let us point out that although the idea was used before for other problems, the adaptation of the procedure to our problem is not trivial at all. Because of the appearance of electromagnetic potential , we must consider our problem for complex-valued functions, and so we need more delicate estimates. Furthermore, we use Lions’ second concentration compactness principle and concentration compactness principle at infinity to prove that the condition holds, which is different from methods used in , .
2 Main results
; , and there is such that the set has finite Lebesgue measure;
() and ;
(h1): and uniformly in x as ;
(h2): there are and such that ;
(h3): there are , and such that and for all , where .
(the bar denotes complex conjugation) this fact means that if , then , and therefore for any .
The spaces and the spaces are not comparable; more precisely, in general and . However, it is proved by Arioli and Szulkin  that if K is a bounded domain with regular boundary, then and are equivalent, where with the norm .
. There exist many functions satisfying condition (H), for example, , where is a positive and bounded function.
Other potentials guaranteeing compactness of the embedding from can also be used in this paper. For example, (1) and ; (2) with periodic function (or bounded function) and .
Recall that we say that a sequence is a sequence at level c (-sequence, for short) if and . is said to satisfy the condition if any -sequence contains a convergent subsequence.
Thus is bounded as . Taking the limit in (3.3) shows that . This completes the proof of Lemma 3.1. □
The main result in this section is the following compactness result.
Suppose that (V), (A) and (H) hold. For any, satisfies thecondition, for all, where; that is, any-sequencehas a strongly convergent subsequence in.
for all , where are Dirac measures at and , are constants.
which leads to a contradiction. Thus, we must have that (II) cannot occur for each i. Thus limit (3.4) holds.
here we use . Thus we prove that strongly converges to u in . This completes the proof of Lemma 3.2. □
4 Proof of Theorem 2.1
In the following, we always consider . By assumptions (V), (A) and (H), one can see that has mountain pass geometry.
Assume that (V), (A) and (H) hold. There existsuch thatifandif, where.
By (2.2) and , we know that the conclusion of Lemma 4.1 holds. This completes the proof of Lemma 4.1. □
for all since all norms in a finite-dimensional space are equivalent and . This completes the proof of Lemma 4.2. □
Since does not satisfy the condition for all , in the following we will find a special finite-dimensional subspace by which we construct sufficiently small minimax levels.
The assumption (V) implies that there is such that . Without loss of generality we assume from now on that .
Then for all , and it suffices to construct small minimax levels for .
Thus we have the following lemma.
for all .
Using this estimate we have the following.
and take . By (4.5), we know that the conclusion of Lemma 4.4 holds. □
We now establish the existence and multiplicity results.
Proof of Theorem 2.1
By Lemma 4.1, we have . In virtue of Lemma 3.2, we know that satisfies the condition, there is such that and , hence the existence is proved.
It follows from Lemma 3.2 that satisfies the condition at all levels . By the usual critical point theory, all are critical levels and has at least pairs of nontrivial critical points. □
The authors would like to appreciate the referees for their precious comments and suggestions about the original manuscript. The authors are supported by the National Natural Science Foundation of China (Grant No. 11301038), Research Foundation during the 12th Five-Year Plan Period of Department of Education of Jilin Province, China (Grant (2013) No. 252), Youth Foundation for Science and Technology Department of Jilin Province (20130522100JH), the open project program of Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University (Grant No. 93K172013K03).
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