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Solutions of perturbed p-Laplacian equation with critical nonlinearity and magnetic fields
Boundary Value Problems volume 2013, Article number: 217 (2013)
In this paper, we consider a perturbed p-Laplacian equation with criticalnonlinearity and magnetic fields on . By using the variational method, we establish theexistence of nontrivial solutions of the least energy.
MSC: 35B33, 35J60, 35J65.
In this paper, we are concerned with the existence of nontrivial solutions to thefollowing 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 Schrödinger equation withmagnetic fields of the form
where ħ 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 ħ 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 agreat charm in physical interest.
Problem (1.3) with has an extensive literature. Different approaches havebeen taken to investigate this problem under various hypotheses on the potential andnonlinearity. See for example [1–18] and the references therein. The above-mentioned papers mostly concentrated onthe nonlinearities with subcritical conditions. Floer and Weinstein in  first studied the existence of single and multiple spike solutions based onthe 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 ofspikes, clustering around a given local maximum point of the potential function. Inaccordance with the Sobolev critical nonlinearities, there have been many papers devotedto studying the existence of solutions to elliptic boundary-valued problems on boundeddomains after the pioneering work by Brézis and Nirenberg . Ding and Lin  first studied the existence of semi-classical solutions to the problem on thewhole space with critical nonlinearities and established the existence of positivesolutions, as well as of those that change sign exactly once. They also obtainedmultiplicity 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 forfixed and special classes of magnetic fields. Cingolani  proved that the magnetic potential only contributes to the phase factor of the solitarysolutions for sufficiently small. For more results, we refer the readerto [19–21] and the references therein.
For general , most of the works studied the existence results toequation (1.1) with . See, for example, [22–28] and the references therein. These papers are mostly devoted to the study ofthe existence of solutions to the problem on bounded domains with the Sobolevsubcritical nonlinearities.
However, to our best knowledge, it seems that there is no work on the existence ofsemiclassical solutions to perturbed p-Laplacian equation on involving critical nonlinearity and magnetic fields. Inthis paper, we consider problem (1.1) with magnetic fields. The main difficulty in thecase 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 mustconsider complex-valued functions for the appearance of electromagnetic potential. To overcome this difficulty, we chiefly follow the ideasof . Notice that although the ideas were used in other problems, the adaption ofthe procedure to our problem is not trivial at all. We need to make careful and complexestimates and prove that the energy functional possesses a Palais-Smale sequence, whichhas 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 1 Assume that (V0), (A0), (K0)and (H1)-(H3) hold. Then forany, there existssuch that if, equation (1.1) has at least one positiveleast 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 thelast section.
Let . Equation (1.1) reads then as
We are going to prove the following result.
Theorem 2 Assume that (V0), (A0), (K0)and (H1)-(H3) are satisfied. Then forany, there existssuch that if, then equation (2.1) has at least onesolution 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 , 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 theproof of Theorem 2.
Lemma 3.1 Let (V0), (A0), (K0)and (H1)-(H3) be satisfied. Forthesequencefor, we get thatandis 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 lossof generality, we may assume that in . Furthermore, passing to a subsequence, we have in for any and a.e. in .
Lemma 3.2 For any, there is a subsequencesuch that for any, there existswith
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 3.3 One has
Proof By direct computation, we easily obtain in . The local compactness of the Sobolev embedding impliesthat, for any , we have
uniformly in . For any , there is such that
for all . By the assumptions and the Hölder inequality, we have
This proof is completed.
Lemma 3.4 One has along a subsequence
By the Brézis-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 3.5 There exists (independent of λ) 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
Denote , then
We obtain the desired conclusion.
Lemma 3.6 There exists a constant (independent of λ) such that ifasequenceforsatisfies, the sequencehas a strongly convergent subsequencein.
Proof By the fact that and Lemma 3.5, we easily get the requiredconclusion.
Now, we consider and prove that the energy functional possesses the mountain pass structure.
Lemma 3.7 Under the assumptions of Theorem 2, thereexistsuch that
Proof The proof of Lemma 3.7 is similar to the one of Lemma 4.1 in .
Lemma 3.8 For 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 λ large enough and small sufficiently, satisfies condition by Lemma 3.6. Furthermore, we will find specialfinite-dimensional subspace, by which we establish sufficiently small minimaxlevels.
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 3.9 Under the assumptions of Theorem 2, forany, there issuch that for each, there existswith, and
whereis 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, thereexists , which satisfies and . That is, is a weak solution of (2.1). Furthermore, it is well knownthat 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 contributed equally in this article. They read and approved the finalmanuscript.
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The authors would like to appreciate the referees for their precious comments andsuggestions about the original manuscript. This research was supported by theNational Natural Science Foundation of China (11271364) and the Fundamental ResearchFunds for the Central Universities (2012QNA46).
The authors declare that they have no competing interests.