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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


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/217


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

© 2013 Zhang and Jiang; licensee Springer.

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 R N . 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

? ? p ? p , A u + V ( x ) | u | p ? 2 u = K ( x ) | u | p ? ? 2 u + f ( x , | u | p ) | u | p ? 2 u , x ? R N , (1.1)

where ? p , A u = div ( | ? u + i A ( x ) u | p ? 2 ( ? u + i A ( x ) u ) ) , i is the imaginary unit, A ( x ) : R N ? R N is a real vector potential, 1 < p < N , p ? = N p / ( N ? p ) denotes the Sobolev critical exponent and N ? 3 .

This paper is motivated by some works concerning the nonlinear Schr?dinger equation with magnetic fields of the form

i ? ? ? ? t = ? ? 2 2 m ( ? + i A ( x ) ) 2 ? + W ( x ) ? ? K ( x ) | ? | 2 ? ? 2 ? ? h ( x , | ? | 2 ) ? for? x ? R N , (1.2)

where ? is Planck?s constant, i is the imaginary unit, 2 ? = 2 N N ? 2 ( N ? 3 ) is the critical exponent, A ( x ) : R N ? R N is a real vector potential, B = curl A and W ( x ) is a scalar electric potential.

In physics, we are interested in the standing wave solutions, that is, solutions to (1.2) of the type

? ( x , t ) = exp ( ? i E t ? ) u ( x ) ,

where ? is a sufficiently small constant, E is a real number, and u ( x ) is a complex-valued function satisfying

? ( ? + i A ( x ) ) 2 u ( x ) + ? ( W ( x ) ? E ) u ( x ) = ? K ( x ) | u | 2 ? ? 2 u + ? h ( x , | u | 2 ) u , x ? R N . (1.3)

We can conduct the transition from quantum mechanics to classical mechanics by letting ? ? 0 . Thus, the existence of semiclassical solutions has a great charm in physical interest.

Problem (1.3) with A ( x ) ? 0 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 A ( x ) ? 0 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 ? > 0 and special classes of magnetic fields. Cingolani [5] proved that the magnetic potential A ( x ) only contributes to the phase factor of the solitary solutions for ? > 0 sufficiently small. For more results, we refer the reader to [19-21] and the references therein.

For general p ? 1 , most of the works studied the existence results to equation (1.1) with A ( x ) ? 0 . 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 R N 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 R N and critical nonlinearity. At the same time, we must consider complex-valued functions for the appearance of electromagnetic potential A ( x ) . 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 V ( x ) , A ( x ) , f ( x , s ) and K ( x ) throughout the paper:

(V0) V ? C ( R N ) , V ( 0 ) = inf x ? R N V ( x ) = 0 , and there exists b > 0 such that the set ? b : = { x ? R N : V ( x ) < b } has a finite Lebesgue measure;

(A0) A ? C ( R N , R N ) and A ( 0 ) = 0 ;

(K0) K ( x ) ? C ( R N , R + ) , 0 < inf K ? sup K < ? ;

(H1) f ? C ( R N ? R + , R ) and f ( x , s ) = o ( | s | ) uniformly in x as s ? 0 ;

(H2) there are c 1 > 0 and p < ? < p ? such that | f ( x , s ) | ? c 1 ( 1 + | s | ? ? p p ) for all ( x , s ) ;

(H3) there exist a 0 > 0 , q > p and ? ? ( p , p ? ) such that F ( x , s ) ? p a 0 | s | q p and ? p F ( x , s ) ? f ( x , s ) s for all ( x , s ) , where F ( x , s ) = ? 0 s f ( x , t ) d t .

Our main result is the following.

Theorem 1Assume that (V0), (A0), (K0) and (H1)-(H3) hold. Then for any ? > 0 , there exists ? ? > 0 such that if ? ? ? ? , equation (1.1) has at least one positive least energy solution u ? , which satisfies

? ? p p ? ? R N ( ? p | ? u ? | p + V ( x ) | u ? | p ) ? ? ? N .

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 ? = ? ? p . Equation (1.1) reads then as

? ? p , A u + ? V ( x ) | u | p ? 2 u = ? K ( x ) | u | p ? ? 2 u + ? f ( x , | u | p ) | u | p ? 2 u , x ? R N . (2.1)

We are going to prove the following result.

Theorem 2Assume that (V0), (A0), (K0) and (H1)-(H3) are satisfied. Then for any ? > 0 , there exists ? ? > 0 such that if ? > ? ? , then equation (2.1) has at least one solution of least energy u ? satisfying

? ? p p ? ? R N ( | ? u ? | p + ? V ( x ) | u ? | p ) ? ? ? 1 ? N p . (2.2)

In order to prove these theorems, we introduce the space

E ? , A = { u ? W 1 , p ( R N , C ) : ? R N ? V ( x ) | u | p < ? , ? > 0 } ,

equipped with the norm

? u ? ? , A = ( ? R N ( | ? u + i ? 1 p A ( x ) u | p + ? V ( x ) | u | p ) ) 1 p .

It is known that E ? , A is the closure of C 0 ? ( R N , C ) . Similar to the diamagnetic inequality [10], we have the following inequality

| ? | u ( x ) | | ? | ? u + i ? 1 p A u | .

In fact, since A ( x ) is real-valued, one has

| ? | u ( x ) | | ? | ? u u ? | u | | = | Re ( ? u + i ? 1 p A u ) u ? | u | | ? | ? u + i ? 1 p A u | (2.3)

(the bar denotes a complex conjugation). This inequality implies that if u ? E ? , A , then | u | ? W 1 , p ( R N ) , and, therefore, u ? L q ( R N ) for any q ? [ p , p ? ) . That is, if u n ? u in E ? , A , then u n ? u in L loc q ( R N ) for any q ? [ p , p ? ) and u n ? u a.e. in R N .

Solutions of (2.1) will be sought in the Sobolev space E ? , A as critical points of the functional

I ? ( u ) = 1 p ? R N ( | ? u + i ? 1 p A u | p + ? V ( x ) | u | p ) ? ? p ? ? R N K ( x ) | u | p ? ? ? p ? R N F ( x , | u | p ) = 1 p ? u ? ? , A p ? ? ? R N G ( x , u ) ,

where G ( x , u ) = 1 p ? K ( x ) | u | p ? + 1 p F ( x , | u | p ) .

It is easy to see that I ? is a C 1 -functional on E ? , A [29].

3 Behavior of ( PS ) c 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 the ( PS ) c sequence { u n } ? E ? , A for I ? , we get that c ? 0 and { u n } is bounded in the space E ? , A .

Proof Under assumptions (K0) and (H3), we have

I ? ( u n ) ? 1 ? I ? ? ( u n ) u n = ( 1 p ? 1 ? ) ? u n ? ? , A p + ( 1 ? ? 1 p ? ) ? ? R N K ( x ) | u n | p ? + ? ? R N ( 1 ? f ( x , | u n | p ) | u n | p ? 1 p F ( x , | u n | p ) ) .

In connection with the facts that I ? ( u n ) ? c and I ? ? ( u n ) ? 0 as n ? ? , we obtain that the ( PS ) c sequence { u n } is bounded in E ? , A , and the energy level c ? 0 .??

Next, let { u n } denote a ( PS ) c sequence. By Lemma?3.1, it is bounded, thus, without loss of generality, we may assume that u n ? u in E ? , A . Furthermore, passing to a subsequence, we have u n ? u in L loc q ( R N ) for any q ? [ p , p ? ) and u n ? u a.e. in R N .

Lemma 3.2For any s ? [ p , p ? ) , there is a subsequence { u n j } such that for any ? > 0 , there exists r ? > 0 with

lim j ? ? sup ? B j ? B r | u n j | s ? ? for any r ? r ? ,

where B r : = { x ? R N : | x | ? r } .

Proof It is easily obtained by the similar proof of Lemma?3.2 [8].??

Let ? ? C ? ( R + ) be a smooth function satisfying 0 ? ? ( t ) ? 1 , ? ( t ) = 1 if t ? 1 and ? ( t ) = 0 if t ? 2 . Define u ? j ( x ) = ? ( 2 | x | / j ) u ( x ) . It is not difficult to see that

? u ? u ? j ? ? , A ? 0 as? j ? ? .

Lemma 3.3One has

lim j ? ? sup Re ? R N ( f ( x , | u n j | p ) | u n j | p ? 2 u n j ? f ( x , u n j ? u ? j ) | u n j ? u ? j | p ? 2 ( u n j ? u ? j ) ? f ( x , | u ? j | p ) | u ? j | p ? 2 u ? j ) ? = 0

uniformly in ? ? E ? , A with ? ? ? ? , A ? 1 .

Proof By direct computation, we easily obtain u ? j ? u in E ? , A . The local compactness of the Sobolev embedding implies that, for any r ? 0 , we have

lim j ? ? sup Re ? B r ( f ( x , | u n j | p ) | u n j | p ? 2 u n j ? f ( x , u n j ? u ? j ) | u n j ? u ? j | p ? 2 ( u n j ? u ? j ) ? f ( x , | u ? j | p ) | u ? j | p ? 2 u ? j ) ? ? = 0

uniformly in ? ? ? ? , A ? 1 . For any ? > 0 , there is r ? ? 0 such that

lim j ? ? sup ? B j ? B r | u ? j | s ? ? R N ? B r | u | s ? ?

for all r ? r ? . By the assumptions and the H?lder inequality, we have

lim j ? ? sup Re ? R N ( f ( x , | u n j | p ) | u n j | p ? 2 u n j ? f ( x , u n j ? u ? j ) | u n j ? u ? j | p ? 2 ( u n j ? u ? j ) ? f ( x , | u ? j | p ) | u ? j | p ? 2 u ? j ) ? = lim j ? ? sup Re ? B j ? B r ( f ( x , | u n j | p ) | u n j | p ? 2 u n j ? f ( x , u n j ? u ? j ) | u n j ? u ? j | p ? 2 ( u n j ? u ? j ) ? f ( x , | u ? j | p ) | u ? j | p ? 2 u ? j ) ? ? c 1 lim j ? ? sup ? B j ? B r ( | u n j | p ? 1 + | u ? j | p ? 1 ) | ? | + c 2 lim j ? ? sup ? B j ? B r ( | u n j | ? ? 1 + | u ? j | ? ? 1 ) | ? | ? c 1 lim j ? ? sup ( ? u n j ? L p p ? 1 ( B j ? B r ) + ? u ? j ? L p p ? 1 ( B j ? B r ) ) ? ? ? L p ( B j ? B r ) + c 2 lim j ? ? sup ( ? u n j ? L ? ( B j ? B r ) ? ? 1 + ? u ? j ? L ? ( B j ? B r ) ? ? 1 ) ? ? ? L ? ( B j ? B r ) ? c 3 ? p ? 1 p + c 4 ? ? ? 1 ? .

This proof is completed.??

Lemma 3.4One has along a subsequence

I ? ( u n ? u ? n ) ? c ? I ? ( u )

and

I ? ? ( u n ? u ? n ) ? 0 in E ? ? 1 ( the dual space of E ? ) .

Proof Combining Lemma?2.1 of [30] and the arguments of [31], one has

I ? ( u n ? u ? n ) = I ? ( u n ) ? I ? ( u ? n ) + ? p ? ? R N K ( x ) ( | u n | p ? ? | u n ? u ? n | p ? ? | u ? n | p ? ) + ? p ? R N ( F ( x , | u n | p ) ? F ( x , | u n ? u ? n | p ) ? F ( x , | u ? n | p ) ) + o ( 1 ) .

By the Br?zis-Lieb lemma [32], we get

lim n ? ? ? R N K ( x ) ( | u n | p ? ? | u n ? u ? n | p ? ? | u ? n | p ? ) = 0

and

lim n ? ? ? R N ( F ( x , | u n | p ) ? F ( x , | u n ? u ? n | p ) ? F ( x , | u ? n | p ) ) = 0 .

We now observe that I ? ( u n ) ? c and I ? ( u ? n ) ? I ? ( u ) , which gives

I ? ( u n ? u ? n ) ? c ? I ? ( u ) .

Moreover, by direct computation, we get

I ? ? ( u n ? u ? n ) ? = I ? ? ( u n ) ? ? I ? ? ( u ? n ) ? + ? Re ? R N K ( x ) ( | u n | p ? ? 2 u n ? | u n ? u ? n | p ? ? 2 ( u n ? u ? n ) ? | u ? n | p ? ? 2 u ? n ) ? + ? Re ? R N ( f ( x , | u n | p ) | u n | p ? 2 u n ? f ( x , | u n ? u ? n | p ) | u n ? u ? n | p ? 2 ( u n ? u ? n ) ? f ( x , | u ? n | p ) | u ? n | p ? 2 u ? n ) ? + o ( 1 ) .

It then follows from the standard arguments that

lim n ? ? Re ? R N K ( x ) ( | u n | p ? ? 2 u n ? | u n ? u ? n | p ? ? 2 ( u n ? u ? n ) ? | u ? n | p ? ? 2 u ? n ) ? = 0

uniformly in ? ? ? ? , A ? 1 . Combining Lemma?3.3, we get I ? ? ( u n ? u ? n ) ? 0 . The proof is completed.??

Let u n 1 = u n ? u ? n , then u n ? u = u n 1 + ( u ? n ? u ) . Therefore, u n ? u in E ? , A if and only if u n 1 ? 0 in E ? , A .

Note that

I ? ( u n 1 ) ? 1 p I ? ? ( u n 1 ) u n 1 = ( 1 p ? 1 p ? ) ? ? R N K ( x ) | u n 1 | p ? + ? ? R N 1 p ( | u n 1 | p f ( x , | u n 1 | p ) ? F ( x , | u n 1 | p ) ) ? ? N ? R N K ( x ) | u n 1 | p ? ? ? N K min ? u n 1 ? p ? p ? ,

where K min = inf x ? R N K ( x ) > 0 . Together with Lemma?3.4, one has

? u n 1 ? p ? p ? ? N ( c ? I ? ( u ) ) ? K min + o ( 1 ) . (3.1)

In the following, we consider the energy level of the functional I ? below which the ( PS ) c condition holds.

Denote V b ( x ) : = max { V ( x ) , b } , where b is the positive constant in assumption (V0). Since the set ? b has a finite measure, combining the fact that u n 1 ? 0 in L loc p ( R N ) , we get

? R N V ( x ) | u n 1 | p = ? R N V b ( x ) | u n 1 | p + o ( 1 ) . (3.2)

Furthermore, by (K0) and (H1)-(H3), there exists C b > 0 such that

? R N ( K ( x ) | u n 1 | p ? + | u n 1 | p f ( x , | u n 1 | p ) ) ? b ? u n 1 ? p p + C b ? u n 1 ? p ? p ? . (3.3)

Let S be the best Sobolev constant of the immersion

S ? u ? p ? p ? ? R N | ? u | p for all? u ? W 1 , p ( R N ) .

Lemma 3.5There exists ? 0 > 0 (independent of?) such that, for any ( PS ) c sequence { u n } ? E ? , A for I ? with u n ? u , either u n ? u in E ? , A or c ? I ? ( u ) ? ? 0 ? 1 ? N p .

Proof Arguing by contradiction, assume that u n ? u , then

lim inf n ? ? ? u n 1 ? ? , A > 0 .

Combining the Sobolev inequality, (3.2) and (3.3), we get

S ? u n 1 ? p ? p ? ? R N | ? u n 1 | p ? ? R N ( | ? u n 1 + i ? 1 p A ( x ) u n 1 | p + ? V ( x ) | u n 1 | p ) ? ? ? R N V ( x ) | u n 1 | p = ? ? R N ( K ( x ) | u n 1 | p ? + | u n 1 | p f ( x , | u n 1 | p ) ) ? ? ? R N V b ( x ) | u n 1 | p + o ( 1 ) ? ? b ? u n 1 ? p p + ? C b ? u n 1 ? p ? p ? ? ? b ? u n 1 ? p p + o ( 1 ) = ? C b ? u n 1 ? p ? p ? + o ( 1 ) ,

which further gives

S ? ? C b ? u n 1 ? p ? p ? ? p + o ( 1 ) ? ? C b ( N ( c ? I ? ( u ) ) ? K min ) p N + o ( 1 ) = ? 1 ? p N C b ( N K min ) p N ( c ? I ? ( u ) ) p N + o ( 1 ) .

Denote ? 0 = S N p C b ? N p N ? 1 K min , then

? 0 ? 1 ? N p ? c ? I ? ( u ) + o ( 1 ) .

We obtain the desired conclusion.??

Lemma 3.6There exists a constant ? 0 > 0 (independent of?) such that if a ( PS ) c sequence { u n } ? E ? , A for I ? satisfies c ? ? 0 ? 1 ? N p , the sequence { u n } has a strongly convergent subsequence in E ? , A .

Proof By the fact that I ? ( u ) ? 0 and Lemma?3.5, we easily get the required conclusion.??

Now, we consider ? ? 1 and prove that the energy functional I ? possesses the mountain pass structure.

Lemma 3.7Under the assumptions of Theorem?2, there exist ? ? , ? ? > 0 such that

I ? ( u ) > 0 if 0 < ? u ? ? , A < ? ? and I ? ( u ) ? ? ? if ? u ? ? , A = ? ? .

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 F ? E ? , A , we have

I ? ( u ) ? ? ? , u ? F as ? u ? ? , A ? ? .

Proof By assumptions (K0) and (H3), one has

I ? ( u ) ? 1 p ? u ? ? , A p ? ? a 0 ? u ? q q for all? u ? E ? , A .

Since all norms in a finite-dimensional space are equivalent, in connection with q > p , we obtain the desired conclusion.??

For ? large enough and c ? small sufficiently, I ? satisfies ( PS ) c ? condition by Lemma?3.6. Furthermore, we will find special finite-dimensional subspace, by which we establish sufficiently small minimax levels.

Define the functional

? ? ( u ) = 1 p ? R N ( | ? u + i ? 1 p A ( x ) u | p + ? V ( x ) | u | p ) ? ? a 0 ? R N | u | q .

It is easy to see that ? ? ? C 1 ( E ? , A ) and I ? ( u ) ? ? ? ( u ) for all u ? E ? , A . Note that

inf { ? R N | ? ? | p : ? ? C 0 ? ( R N , R ) , ? ? ? L q ( R N ) = 1 } = 0 .

For any ? > 0 , there is ? ? ? C 0 ? ( R N , R ) with ? ? ? ? L q ( R N ) = 1 and supp ? ? ? B r ? ( 0 ) such that ? ? ? ? ? p p < ? . Let e ? ( x ) = ? ? ( ? p x ) , then supp e ? ? B ? ? 1 p r ? ( 0 ) . For any t ? 0 , we have

? ? ( t e ? ) = t p p ? e ? ? ? , A p ? a 0 ? t q ? R N | ? ? ( ? p x ) | q = ? 1 ? N p J ? ( t ? ? ) ,

where

J ? ( u ) = 1 p ? R N ( | ? u + i ? 1 p A ( x ) u | p + V ( ? ? 1 p x ) | u | p ) ? a 0 ? R N | u | q .

We derive that

max t ? 0 J ? ( t ? ? ) ? q ? p p q ( q a 0 ) p q ? p ( ? R N ( | ? ? ? | p + ( A ( ? ? 1 p x ) + V ( ? ? 1 p x ) ) | ? ? | p ) ) q q ? p .

Observe that A ( 0 ) = 0 , V ( 0 ) = 0 and ? ? ? ? ? p p < ? . Therefore, there exists ? ? > 0 such that for all ? ? ? ? , we have

max t ? 0 I ? ( t ? ? ) ? ( q ? p p q ( q a 0 ) p q ? p ( 5 ? ) q q ? p ) ? 1 ? N p . (3.4)

Lemma 3.9Under the assumptions of Theorem?2, for any ? > 0 , there is ? ? > 0 such that for each ? ? ? ? , there exists e ? ? ? E ? , A with ? e ? ? ? ? , A > ? ? , I ? ( e ? ? ) ? 0 and

max t ? 0 I ? ( t e ? ? ) ? ? ? 1 ? N p ,

where ? ? is defined in Lemma?3.7.

Proof For any ? > 0 , we can choose ? < 0 so small that

q ? p p q ( q a 0 ) p q ? p ( 5 ? ) q q ? p ? ? .

Denote e ? ( x ) = ? ? ( ? p x ) and ? ? = ? ? . Let t ? ? > 0 be such that t ? ? ? e ? ? ? , A > ? ? and I ? ( t e ? ) ? 0 for all t ? t ? ? . Then, combining (3.4), e ? ? = t ? ? e ? 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 ? > 0 with 0 < ? < ? 0 , we choose ? ? > 0 and define the minimax value

c ? = inf ? ? ? ? max t ? [ 0 , 1 ] I ? ( ? ( t ) ) with? c ? ? ? ? 1 ? N p ?for each? ? ? ? ? ,

where ? ? = { ? ? C ( [ 0 , 1 ] , E ? , A ) : ? ( 0 ) = 0 , ? ( 1 ) = e ? ? } .

Lemma?3.6 shows that I ? satisfies ( PS ) c ? condition. Therefore, by the mountain pass theorem, there exists u ? ? E ? , A , which satisfies I ? ( u ? ) = c ? and I ? ? ( u ? ) = 0 . That is, u ? is a weak solution of (2.1). Furthermore, it is well known that u ? is the least energy solution of equation (2.1).

Moreover, together with I ? ( u ? ) ? ? ? 1 ? N p and I ? ? ( u ? ) = 0 , we have

I ? ( u ? ) = I ? ( u ? ) ? 1 ? I ? ? ( u ? ) ( u ? ) = ( 1 p ? 1 ? ) ? u ? ? ? , A p + ( 1 ? ? 1 p ? ) ? ? R N K ( x ) | u ? | p ? + ? ? R N ( 1 ? | u ? | p f ( x , | u ? | p ) ? 1 p F ( x , | u ? | p ) ) ? ( 1 p ? 1 ? ) ? u ? ? ? , A p .

By inequality (2.3), we obtain

? ? p p ? ? R N ( | ? u ? | p + ? V ( x ) | u ? | p ) ? ? ? 1 ? N p .

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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