- Open Access
Solutions of perturbed p-Laplacian equation with critical nonlinearity and magnetic fields
© Zhang and Jiang; licensee Springer. 2013
- Received: 3 April 2013
- Accepted: 28 August 2013
- Published: 5 November 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.
- p-Laplacian equation
- critical nonlinearity
- magnetic fields
- mountain pass theorem
where , i is the imaginary unit, is a real vector potential, , denotes the Sobolev critical exponent and.
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.
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.
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.
We are going to prove the following result.
(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 .
It is easy to see that is a -functional on .
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.
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 .
Proof It is easily obtained by the similar proof of Lemma 3.2 .
This proof is completed.
uniformly in . Combining Lemma 3.3, we get . The proof is completed.
Let , then . Therefore, in if and only if in .
In the following, we consider the energy level of the functional below which the condition holds.
Lemma 3.5 There exists (independent of λ) such that,for anysequenceforwith, eitherinor.
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.
Proof The proof of Lemma 3.7 is similar to the one of Lemma 4.1 in .
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.
whereis defined in Lemma 3.7.
Denote and . Let be such that and for all . Then, combining (3.4), meets the requirements.
In this section, we give the proof of Theorem 2.
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).
The proof is complete.
The authors contributed equally in this article. They read and approved the finalmanuscript.
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).
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