# Solutions of perturbed p-Laplacian equation with critical nonlinearity and magnetic fields

Huixing Zhang* and Juan Jiang

Author Affiliations

Department of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu, 221116, People?s Republic of China

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Boundary Value Problems 2013, 2013:217  doi:10.1186/1687-2770-2013-217

 Received: 3 April 2013 Accepted: 28 August 2013 Published: 5 November 2013

This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

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

### 1 Introduction

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

(1.1)

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

(1.2)

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

(1.3)

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 [11] 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 [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 boundary-valued problems on bounded domains after the pioneering work by Br?zis and Nirenberg [4]. Ding and Lin [8] 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 [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 [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 [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 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 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

(2.1)

We are going to prove the following result.

Theorem 2Assume that (V0), (A0), (K0) and (H1)-(H3) are satisfied. Then for any, there existssuch that if, then equation (2.1) has at least one solution of least energysatisfying

(2.2)

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 real-valued, one has

(2.3)

(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

where .

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 (V0), (A0), (K0) and (H1)-(H3) be satisfied. For thesequencefor, 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 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

where.

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

uniformly inwith.

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?zis-Lieb 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

(3.1)

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

(3.2)

Furthermore, by (K0) and (H1)-(H3), there exists such that

(3.3)

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

Denote , then

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 (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

where

We derive that

Observe that , and . Therefore, there exists such that for all , we have

(3.4)

Lemma 3.9Under the assumptions of Theorem?2, for any, 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

where .

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.

### 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).

### References

1. Ambrosetti, A, Rabinowitz, PH: Dual variational methods in critical point theory and applications. J. Funct. Anal.. 14, 349–381 (1973). Publisher Full Text

2. Bartsch, T, Dancer, E, Peng, S: On multi-bump semi-classical bound states of nonlinear Schr?dinger equations with electromagnetic fields. Adv. Differ. Equ.. 11, 781–812 (2006)

3. Benci, V: On critical point theory of indefinite functions in the presence of symmetries. Trans. Am. Math. Soc.. 274, 533–572 (1982). Publisher Full Text

4. Br?zis, H, Nirenberg, L: Positive solutions of nonlinear elliptic equation involving critical Sobolev exponents. Commun. Pure Appl. Math.. 16, 437–477 (1983)

5. Cingolani, S: Semiclassical stationary states of nonlinear Schr?dinger equation with an external magnetic field. J.?Differ. Equ.. 188, 52–79 (2003). Publisher Full Text

6. Clapp, M, Ding, YH: Minimal nodal solutions of a Schr?dinger equation with critical nonlinearity and symmetric potential. Differ. Integral Equ.. 16, 981–992 (2003)

7. Del Pino, M, Felmer, PL: Semi-classical states for nonlinear Schr?dinger equations. J. Funct. Anal.. 149, 245–265 (1997). Publisher Full Text

8. Ding, YH, Lin, FH: Solutions of perturbed Schr?dinger equations with critical nonlinearity. Calc. Var.. 30, 231–249 (2007). Publisher Full Text

9. Ding, YH, Szulkin, A: Bound states for semilinear Schr?dinger equations with sign-changing potential. Calc. Var. Partial Differ. Equ.. 29, 397–419 (2007). Publisher Full Text

10. Esteban, M, Lions, PL: Stationary solutions of nonlinear Schr?dinger equation with an external magnetic field. PDE and Calculus of Variations: In Honor of E De Giorgi, pp. 369–408. Birkh?user, Basel (1990)

11. Floer, A, Weinstein, A: Nonspreading wave packets for the cubic Schr?dinger equation with a bounded potential. J.?Funct. Anal.. 69, 397–408 (1986). Publisher Full Text

12. Grossi, M: Some results on a class of nonlinear Schr?dinger equations. Math. Z.. 235, 687–705 (2000). Publisher Full Text

13. Jeanjean, L, Tanaka, K: Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Calc. Var. Partial Differ. Equ.. 21, 287–318 (2004)

14. Kang, X, Wei, J: On interacting bumps of semi-classical states of nonlinear Schr?dinger equations. Adv. Differ. Equ.. 5, 899–928 (2000)

15. Li, YY: On a singularly perturbed elliptic equation. Adv. Differ. Equ.. 2, 955–980 (1997)

16. Oh, YG: Existence of semiclassical bound states of nonlinear Schr?dinger equations with potentials of the class . Commun. Partial Differ. Equ.. 13, 1499–1519 (1988). Publisher Full Text

17. Oh, YG: On positive multi-lump bound states of nonlinear Schr?dinger equations under multiple well potential. Commun. Math. Phys.. 131, 223–253 (1990). Publisher Full Text

18. Pistoia, A: Multi-peak solutions for a class of some results on a class of nonlinear Schr?dinger equations. Nonlinear Differ. Equ. Appl.. 9, 69–91 (2002). Publisher Full Text

19. Tang, Z: On the least energy solutions of nonlinear Schr?dinger equations with electromagnetic fields. Comput. Math. Appl.. 54, 627–637 (2007). Publisher Full Text

20. Tang, Z: Multi-bump bound states of nonlinear Schr?dinger equations with electromagnetic fields and critical frequency. J. Differ. Equ.. 245, 2723–2748 (2008). Publisher Full Text

21. Wang, F: On an electromagnetic Schr?dinger equation with critical growth. Nonlinear Anal.. 69, 4088–4098 (2008). Publisher Full Text

22. Alves, CO, Ding, YH: Multiplicity of positive solutions to a p-Laplacian equation involving critical nonlinearity. J. Math. Anal. Appl.. 279, 508–521 (2003). Publisher Full Text

23. El Khalil, A, El Manouni, S, Ouanan, M: On some nonlinear elliptic problems for p-Laplacian in . NoDEA Nonlinear Differ. Equ. Appl.. 15, 295–307 (2008). Publisher Full Text

24. Fan, XL: -Laplacian equations in with periodic data and nonperiodic perturbations. J. Math. Anal. Appl.. 341, 103–119 (2008). Publisher Full Text

25. Habib, SE, Tsouli, N: On the spectrum of the p-Laplacian operator for Neumann eigenvalue problems with weights. Electron. J. Differ. Equ. Conf.. 14, 181–190 (2005)

26. L?, A: Eigenvalue problems for the p-Laplacian. Nonlinear Anal.. 64, 1057–1099 (2006). Publisher Full Text

27. Liu, CG, Zheng, YQ: Existence of nontrivial solutions for p-Laplacian equations in . J. Math. Anal. Appl.. 380, 669–679 (2011). Publisher Full Text

28. Man?sevich, R, Mawhin, J: Boundary value problems for nonlinear perturbations of vector p-Laplacian-like operators. J. Korean Math. Soc.. 5, 665–685 (2000)

29. Mawhin, J, Willem, M: Critical Point Theory and Hamiltonian Systems, Springer, New York (1989)

30. Li, YY, Guo, QQ, Niu, PC: Global compactness results for quasilinear elliptic problems with combined critical Sobolev-Hardy terms. Nonlinear Anal.. 74, 1445–1464 (2011). Publisher Full Text

31. Ghoussoub, N, Yuan, C: Multiple solutions for quasi-linear PDES involving the critical Sobolev and Hardy exponents. Trans. Am. Math. Soc.. 352, 5703–5743 (2000). Publisher Full Text

32. Br?zis, H, Lieb, E: A relation between pointwise convergence of functions and convergence of functional. Proc. Am. Math. Soc.. 88, 486–490 (1983). Publisher Full Text