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Existence of nontrivial solutions for perturbed p-Laplacian system involving critical nonlinearity and magnetic fields

Huixing Zhang*, Jiaying Liu and Wenbin Liu

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

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Citation and License

Boundary Value Problems 2013, 2013:11  doi:10.1186/1687-2770-2013-11


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/11


Received:5 October 2012
Accepted:9 January 2013
Published:24 January 2013

© 2013 Zhang et al.; 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

Under the suitable assumptions, we establish the existence of nontrivial solutions for a perturbed p-Laplacian system in R N with critical nonlinearity and magnetic fields by using the variational method.

MSC: 35B33, 35J60, 35J65.

Keywords:
p-Laplacian system; critical nonlinearity; magnetic fields; variational method

1 Introduction

In this paper, we consider a class of quasi-linear elliptic systems of the form

{ ε p Δ p , A u + V ( x ) | u | p 2 u = H s ( | u | p , | v | p ) | u | p 2 u + K ( x ) | u | p 2 u , x R N , ε p Δ p , A v + V ( x ) | v | p 2 v = H t ( | u | p , | v | p ) | v | p 2 v + K ( x ) | v | p 2 v , x R N , (1.1)

where Δ 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 real vector potential, 1 < p < N , V ( x ) is a non-negative potential, p = N p / ( N p ) denotes the Sobolev critical exponent for N 3 and K ( x ) is a bounded positive coefficient.

The scalar case corresponding to (1.1) has received considerable attention in recent years. For p = 2 and A ( x ) 0 , the scalar case corresponding to (1.1) turns into

ε 2 Δ u + V ( x ) u = K ( x ) | u | 2 2 u + f ( x , | u | 2 ) u , x R N . (1.2)

The equation (1.2) arises in finding standing wave solutions of the nonlinear Schrödinger equation

i ħ ψ t = ħ 2 2 m Δ ψ + W ( x ) ψ g ( x , | ψ | ) ψ . (1.3)

A standing wave solution of (1.3) is a solution of the form

ψ ( x , t ) = u ( x ) exp i E t ħ .

Then ψ ( x , t ) solves (1.3) if and only if u ( x ) solves (1.2) with V ( x ) = W ( x ) E and ε 2 = ħ 2 2 m .

The equation (1.2) has been extensively investigated in the literature based on various assumptions of the potential V ( x ) and the nonlinearity f ( x , u ) . See, for example, [1-15] and the references therein.

There are also many works dealing with the magnetic fields A ( x ) 0 and p = 2 for the scalar case corresponding to (1.1). In [16], the authors firstly obtained the existence of standing waves for special classes of magnetic fields. For many results, we refer the reader to [17-22].

For general p > 1 , most of the work, as we know, consider the scalar case which corresponds to (1.1) with A ( x ) 0 . See [23-27] and the references therein. We especially mention [24] for the existence of positive solutions for a class of p-Laplacian equations. Gloss [24] studied the existence and asymptotic behavior of positive solutions for quasi-linear elliptic equations of the form

ε p Δ p u + V ( x ) | u | p 2 u = f ( u ) , x R N , (1.4)

where f is a subcritical nonlinearity without some growth conditions such as the Ambrosetti-Rabinowitz condition. The problem (1.4) has also been studied in [28-32]. The main difficulty in treating this class of equation (1.4) is a possible lack of compactness due to the unboundedness of the domain.

However, to our best knowledge, it seems there is almost no work on the existence of non-trivial solutions to the problem (1.1) involving critical nonlinearity and magnetic fields. We mainly follow the idea of [7]. Observe that though the idea was used in other problems, the adaption of the procedure to the problem is not trivial at all. Because of the appearance of magnetic fields A ( x ) , we must deal with the problem for complex-valued functions and therefore we need more delicate estimates.

The outline of the paper is as follows. The forthcoming section is the main result and preliminary results including the appropriate space setting to work with. In Section 3, we study the behavior of ( PS ) c sequence. Section 4 gets that the functional associated to the problem possesses the mountain geometry structure, and the last section concludes the proof of the main result.

2 Main results and preliminaries

Firstly, we make the following assumptions on V ( x ) , A ( x ) , H ( s , t ) and K ( x ) throughout the paper:

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

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

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

( H 1 ) H ( s , t ) C 1 ( R + × R + , R ) and H s , H t = o ( | s | + | t | ) as | s | + | t | 0 ;

( H 2 ) there exist c 1 > 0 and p < α < p such that

| H s ( s , t ) | , | H t ( s , t ) | c 1 ( 1 + | s | α p p + | t | α p p ) ;

( H 3 ) there are a 0 > 0 , θ ( p , p ) and α , β > p such that H ( s , t ) p a 0 ( | s | α p + | t | β p ) and 0 < θ p H ( s , t ) s H s + t H t .

Under the above mentioned conditions, we get the following result.

Theorem 1Suppose that the assumptions ( V 0 ), ( A 0 ), ( K 0 ) and ( H 1 )-( H 3 ) hold. Then for any σ > 0 , there is ε σ > 0 such that if ε < ε σ , the problem (1.1) has at least one solution ( u ε , v ε ) which satisfies

θ p p θ R N ( ε p ( | | u ε | | p + | | v ε | | p ) + V ( x ) | u ε | p + V ( x ) | v ε | p ) σ ε N .

Setting λ = ε p , the problem (1.1) is equivalent to the following problem:

{ Δ p , A u + λ V ( x ) | u | p 2 u = λ H s ( | u | p , | v | p ) | u | p 2 u + λ K ( x ) | u | p 2 u , x R N , Δ p , A v + λ V ( x ) | v | p 2 v = λ H t ( | u | p , | v | p ) | v | p 2 v + λ K ( x ) | v | p 2 v , x R N . (2.1)

We are going to prove the following result.

Theorem 2Suppose that the assumptions ( V 0 ), ( A 0 ), ( K 0 ) and ( H 1 )-( H 3 ) hold. Then for any σ > 0 , there is Λ σ > 0 such that if λ > Λ σ , the problem (2.1) has at least one solution ( u λ , v λ ) which satisfies

θ p p θ R N ( ( | | u λ | | p + | | v λ | | p ) + λ V ( x ) | u λ | p + λ V ( x ) | v λ | p ) σ λ 1 p N . (2.2)

For convenience, we quote the following notations. Let E λ , A denote the Banach space

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

equipped with the norm

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

Set E = E λ , A × E λ , A and ( u , v ) E p = u λ , A p + v λ , A p for any ( u , v ) E .

Similar to the diamagnetic inequality [16], we have

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

(the bar denotes complex conjugation). This inequality shows that if u E λ , A , then | u | W 1 , p ( R N ) and therefore u L q ( R N ) for any q [ p , p ) . That is to say, if u n u in E λ , 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 .

The energy functional associated with (2.1) is defined by

I λ ( u , v ) = 1 p R N ( | u + i λ 1 p A u | p + λ V ( x ) | u | p + | v + i λ 1 p A v | p + λ V ( x ) | v | p ) λ p R N K ( x ) ( | u | p + | v | p ) λ p R N H ( | u | p , | v | p ) = 1 p ( u , v ) E p λ R N G ( u , v ) ,

where G ( u , v ) = 1 p K ( x ) ( | u | p + | v | p ) + 1 p H ( | u | p , | v | p ) .

Under the assumptions of Theorem 2, standard arguments [33] show that I λ C 1 ( E λ , A , R ) and its critical points are weak solutions of the equation (2.1).

3 ( PS ) c condition

We call a sequence { ( u n , v n ) } E a ( PS ) c sequence if I λ ( u n , v n ) c and I λ ( u n , v n ) 0 strongly in E ( E is the dual space of E). I λ is said to satisfy the ( PS ) c condition if any ( PS ) c sequence contains a convergent subsequence.

The main result of Section 3 is the following compactness result.

Proposition 3.1Let the assumptions of Theorem 2 be satisfied. There exists a constant α 0 0 independent ofλsuch that, for any ( PS ) c sequence { ( u n , v n ) } E for I λ with ( u n , v n ) ( u , v ) , either ( u n , v n ) ( u , v ) or c I λ ( u , v ) α 0 λ 1 N p .

As a consequence, we obtain the following result.

Proposition 3.2Assume that the assumptions of Proposition 3.1 hold, I λ ( u , v ) satisfies the ( PS ) c condition for all c α 0 λ 1 N p .

In order to prove Proposition 3.1, we need the following lemmas.

Lemma 3.1Let the assumptions of Theorem 2 be satisfied. { ( u n , v n ) } is a ( PS ) c sequence of  I λ . Then c 0 and { ( u n , v n ) } is bounded in the spaceE.

Proof

One has

I λ ( u n , v n ) 1 θ I λ ( u n , v n ) ( u n , v n ) = ( 1 p 1 θ ) ( u n , v n ) E p + ( 1 θ 1 p ) λ R N K ( x ) ( | u n | p + | v n | p ) + λ R N ( 1 θ ( | u n | p H s ( | u n | p , | v n | p ) + | v n | p H t ( | u n | p , | v n | p ) ) 1 p H ( | u n | p , | v n | p ) ) ( 1 p 1 θ ) ( u n , v n ) E p .

Together with I λ ( u n , v n ) c and I λ ( u n , v n ) 0 as n , we have

( 1 p 1 θ ) ( u n , v n ) E p c + o ( 1 ) + ε n ( u n , v n ) E .

Then { ( u n , v n ) } is bounded and c 0 . □

From Lemma 3.1, we may assume ( u n , v n ) ( u , v ) in E and ( u n , v n ) ( u , v ) in L loc q ( R N ) × L loc q ( R N ) for any q [ p , p ) and u n u , v n v a.e. in R N .

Lemma 3.2Let γ [ p , p ) . There is a subsequence { ( u n j , v n j ) } such that for any ε > 0 , there is r ε > 0 with r r ϵ

lim j sup B j B r ( | u n j | γ + | v n j | γ ) ε ,

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

Proof The proof of Lemma 3.2 is similar to that of Lemma 3.2 of [27], so we omit it. □

Let η C ( R + ) be a smooth function satisfying 0 η ( t ) 1 , η ( t ) = 1 if t 1 and η ( t ) = 0 if t 2 . Define u ˜ j ( x ) = η ( 2 | x | / j ) u ( x ) , v ˜ j ( x ) = η ( 2 | x | / j ) v ( x ) . Obviously, we have

u u ˜ j E λ , A 0 and v v ˜ j E λ , A 0 as  j . (3.1)

Lemma 3.3One has

lim j Re R N ( H s ( | u n j | p , | v n j | p ) | u n j | p 2 u n j H s ( | u n j u ˜ j | p , | v n j v ˜ j | p ) | u n j u ˜ j | p 2 ( u n j u ˜ j ) H s ( | u ˜ j | p , | v ˜ j | p ) | u ˜ j | p 2 u ˜ j ) φ ¯ = 0

and

lim j Re R N ( H t ( | u n j | p , | v n j | p ) | v n j | p 2 v n j H t ( | u n j u ˜ j | p , | v n j v ˜ j | p ) | v n j v ˜ j | p 2 ( v n j v ˜ j ) H t ( | u ˜ j | p , | v ˜ j | p ) | v ˜ j | p 2 v ˜ j ) ψ ¯ = 0

uniformly in ( φ , ψ ) E with ( φ , ψ ) E 1 .

Proof The local compactness of Sobolev embedding implies that for any r 0 , we have

lim j Re B r ( H s ( | u n j | p , | v n j | p ) | u n j | p 2 u n j H s ( | u n j u ˜ j | p , | v n j v ˜ j | p ) | u n j u ˜ j | p 2 ( u n j u ˜ j ) H s ( | u ˜ j | p , | v ˜ j | p ) | u ˜ j | p 2 u ˜ j ) φ ¯ = 0

uniformly in φ E λ , A 1 . For any ε > 0 , there exists r ε > 0 such that

lim j sup B j B r | u ˜ j | γ R N | u | γ ε

for all r r ϵ . Together with the assumption ( H 2 ) and the Hölder inequality, it follows from Lemma 3.2 that

lim j sup Re R N ( H s ( | u n j | p , | v n j | p ) | u n j | p 2 u n j H s ( | u n j u ˜ j | p , | v n j v ˜ j | p ) | u n j u ˜ j | p 2 ( u n j u ˜ j ) H s ( | u ˜ j | p , | v ˜ j | p ) | u ˜ j | p 2 u ˜ j ) φ ¯ = lim j sup Re B j B r ( H s ( | u n j | p , | v n j | p ) | u n j | p 2 u n j H s ( | u n j u ˜ j | p , | v n j v ˜ j | p ) | u n j u ˜ j | p 2 ( u n j u ˜ j ) H s ( | u ˜ j | p , | v ˜ j | p ) | u ˜ j | p 2 u ˜ j ) φ ¯ c 1 lim j sup B j B r ( | u n j | p 1 + | v n j | p 1 + | u ˜ j | p 1 + | v ˜ j | p 1 ) | φ ¯ | + c 2 lim j sup B j B r ( | u n j | α 1 + | v n j | α 1 + | u ˜ j | α 1 + | v ˜ j | α 1 ) | φ ¯ | c 1 lim j sup ( u n j L p ( B j B r ) p 1 + v n j L p ( B j B r ) p 1 + u ˜ j L p ( B j B r ) p 1 + v ˜ j L p ( B j B r ) p 1 ) φ ¯ L p ( B j B r ) + c 2 lim j sup ( u n j L α ( B j B r ) α 1 + v n j L α ( B j B r ) α 1 + u ˜ j L α ( B j B r ) α 1 + v ˜ j L α ( B j B r ) α 1 ) φ ¯ L α ( B j B r ) c 3 ε p 1 p + c 4 ε α 1 α ,

where c i ( i = 1 , 2 , 3 , 4 ) are positive constants. Similarly, we can prove

lim j Re R N ( H t ( | u n j | p , | v n j | p ) | v n j | p 2 v n j H t ( | u n j u ˜ j | p , | v n j v ˜ j | p ) | v n j v ˜ j | p 2 ( v n j v ˜ j ) H t ( | u ˜ j | p , | v ˜ j | p ) | v ˜ j | p 2 v ˜ j ) ψ ¯ = 0 .

 □

Lemma 3.4Let { ( u n , v n ) } and { ( u ˜ n , v ˜ n ) } be as defined above. Then the following conclusions hold:

I λ ( u n u ˜ n , v n v ˜ n ) c I λ ( u , v )

and

I λ ( u n u ˜ n , v n v ˜ n ) 0 in  E ( the dual space of  E ) .

Proof By using the similar arguments of [34,35], we have

I λ ( u n u ˜ n , v n v ˜ n ) = I λ ( u n , v n ) I λ ( u ˜ n , v ˜ n ) + λ p R N K ( x ) ( ( | u n | p | u n u ˜ n | p | u ˜ n | p ) + ( | v n | p | v n v ˜ n | p | v ˜ n | p ) ) + λ p R N ( H ( | u n | p , | v n | p ) H ( | u n u ˜ n | p , | v n v ˜ n | p ) H ( | u ˜ n | p , | v ˜ n | p ) ) + o ( 1 ) .

By (3.1) and the similar idea of proving the Brézis-Lieb lemma [36], it is easy to get

lim n R N K ( x ) ( ( | u n | p | u n u ˜ n | p | u ˜ n | p ) + ( | v n | p | v n v ˜ n | p | v ˜ n | p ) ) = 0

and

lim n R N ( H ( | u n | p , | v n | p ) H ( | u n u ˜ n | p , | v n v ˜ n | p ) H ( | u ˜ n | p , | v ˜ n | p ) ) = 0 .

Furthermore, using the fact I λ ( u n , v n ) c and I λ ( u ˜ n , v ˜ n ) I λ ( u , v ) , we obtain

I λ ( u n u ˜ n , v n v ˜ n ) c I λ ( u , v ) .

In order to prove I λ ( u n u ˜ n , v n v ˜ n ) 0 in E 1 , for any ( φ , ψ ) E , it follows that

I λ ( u n u ˜ n , v n v ˜ n ) ( φ , ψ ) = I λ ( u n , v n ) ( φ , ψ ) I λ ( u ˜ n , v ˜ n ) ( φ , ψ ) + λ 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 ) φ ¯ + λ Re R N K ( x ) ( | v n | p 2 v n | v n v ˜ n | p 2 ( v n v ˜ n ) | v ˜ n | p 2 v ˜ n ) ψ ¯ + λ Re R N ( H s ( | u n | p , | v n | p ) | u n | p 2 u n H s ( | u n u ˜ n | p , | v n v ˜ n | p ) | u n u ˜ n | p 2 ( u n u ˜ n ) H s ( | u ˜ n | p , | v ˜ n | p ) | u ˜ n | p 2 u ˜ n ) φ ¯ + λ Re R N ( H t ( | u n | p , | v n | p ) | v n | p 2 v n H t ( | u n u ˜ n | p , | v n v ˜ n | p ) | v n v ˜ n | p 2 ( v n v ˜ n ) H t ( | u ˜ n | p , | v ˜ n | p ) | v ˜ n | p 2 v ˜ n ) ψ ¯ + o ( 1 ) .

It is standard to check that

lim n 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 ) φ ¯ = 0

and

lim n R N K ( x ) ( | v n | p 2 v n | v n v ˜ n | p 2 ( v n v ˜ n ) | v ˜ n | p 2 v ˜ n ) ψ ¯ = 0

uniformly in ( φ , ψ ) E with ( φ , ψ ) E 1 . Together with Lemma 3.3, we have

I λ ( u n u ˜ n , v n v ˜ n ) 0 in  E .

 □

Let u n 1 = u n u ˜ n , v n 1 = v n v ˜ n , then u n u = u n 1 + ( u ˜ n u ) , v n v = v n 1 + ( v ˜ n v ) . From (3.1), we get ( u n , v n ) ( u , v ) in E if and only if ( u n 1 , v n 1 ) ( 0 , 0 ) in E.

Observe that

I λ ( u n 1 , v n 1 ) 1 p I λ ( u n 1 , v n 1 ) ( u n 1 , v n 1 ) = ( 1 p 1 p ) λ R N K ( x ) ( | u n 1 | p + | v n 1 | p ) + λ p R N ( | u n 1 | p H s ( | u n 1 | p , | v n 1 | p ) + | v n 1 | p H t ( | u n 1 | p , | v n 1 | p ) H ( | u n 1 | p , | v n 1 | p ) ) λ N R N K ( x ) ( | u n 1 | p + | v n 1 | p ) λ N K min R N ( | u n 1 | p + | v n 1 | p ) ,

where K min = inf x R N K ( x ) > 0 . Furthermore, we get

( u n 1 , v n 1 ) p p N ( c I λ ( u , v ) ) λ K min + o ( 1 ) . (3.2)

Now, we consider the energy level of the functional I λ below which the ( PS ) c condition holds.

Let V b ( x ) : = max { V ( x ) , b } , where b is a positive constant in the assumption ( V 0 ). Since the set ν b has finite measure, we get

R N V ( x ) ( | u n 1 | p + | v n 1 | p ) = R N V b ( x ) ( | u n 1 | p + | v n 1 | p ) + o ( 1 ) . (3.3)

In connection with the assumptions ( H 1 )-( H 3 ) and the Young inequality, there exists C b > 0 such that

(3.4)

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 ) .

Proof of Proposition 3.1 Assume that ( u n , v n ) ( u , v ) , then

lim inf n ( u n 1 , v n 1 ) E > 0

and

c I λ ( u , v ) > 0 .

By the Sobolev embedding inequality and the diamagnetic inequality, we get

S ( u n 1 p p + v n 1 p p ) R N ( | u n 1 | p + | v n 1 | p ) R N ( | u n 1 + i λ 1 p A ( x ) u n 1 | p + | v n 1 + i λ 1 p A ( x ) v n 1 | p ) = R N ( | u n 1 + i λ 1 p A ( x ) u n 1 | p + λ V ( x ) | u n 1 | p + | v n 1 + i λ 1 p A ( x ) v n 1 | p + λ V ( x ) | v n 1 | p ) λ R N V ( x ) ( | u n 1 | p + | v n 1 | p ) = λ R N K ( x ) ( | u n 1 | p + | v n 1 | p ) + | u n 1 | p H s ( | u n 1 | p , | v n 1 | p ) + | v n 1 | p H t ( | u n 1 | p , | v n 1 | p ) λ R N V b ( x ) ( | u n 1 | p + | v n 1 | p ) + o ( 1 ) λ b ( u n 1 p p + v n 1 p p ) + λ C b ( u n 1 p p + v n 1 p p ) λ b ( u n 1 p p + v n 1 p p ) + o ( 1 ) = λ C b ( u n 1 p p + v n 1 p p ) + o ( 1 ) .

This, together with (3.2), gives

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

Set α 0 = S N p C b N p N 1 K min , then

α 0 λ 1 N p c I λ ( u , v ) + o ( 1 ) .

This completes the proof of Proposition 3.1. □

Proof of Proposition 3.2 Since c α 0 λ 1 N p , we have

c I λ ( u , v ) α 0 λ 1 N p I λ ( u , v ) .

In connection with I λ ( u , v ) 0 and Proposition 3.1, we complete this proof. □

4 The mountain-pass structure

In the following, we always consider λ 1 . We will prove that I λ possesses the mountain-pass structure which has been carefully discussed in the works [37,38].

Lemma 4.1Let the assumptions of Theorem 2 be satisfied. There exist α λ , ρ λ > 0 such that

I λ ( u , v ) > 0 if  0 < ( u , v ) E < ρ λ and I λ ( u , v ) α λ if  ( u , v ) E = ρ λ .

Proof By (3.4), for any δ > 0 , there is C δ > 0 such that

R N G ( u , v ) δ ( u p p + v p p ) + C δ ( u p p + v p p ) .

Thus,

I λ ( u , v ) = 1 p ( u , v ) E p λ R N G ( u , v ) 1 p ( u , v ) E p λ δ ( u p p + v p p ) λ C δ ( u p p + v p p ) .

In connection with u p p + v p p C 1 ( u , v ) E p , we may choose δ ( 2 p λ C 1 ) 1 such that

I λ ( u , v ) 1 2 p ( u , v ) E p λ C δ ( u p p + v p p ) .

The fact p > p implies the desired conclusion. □

Lemma 4.2Under the assumptions of Lemma 4.1, for any finite dimensional subspace F E , we have

I λ ( u , v ) as  ( u , v ) F , ( u , v ) E .

Proof Together with the fact H ( s , t ) p a 0 ( | s | α p + | t | β p ) , we have

I λ ( u , v ) 1 p ( u , v ) E p λ a 0 ( u α α + v β β ) for all  ( u , v ) E .

Since all norms in a finite-dimensional space are equivalent and α , β > p , we complete the proof. □

In the following, we will find special finite-dimensional subspaces by which we establish sufficiently small mini-max levels.

Define the functional

Φ λ ( u , v ) = 1 p ( u , v ) E p λ a 0 R N ( | u | α + | v | β ) .

Obviously, it follows that Φ λ C 1 ( E ) and I λ ( u , v ) Φ λ ( u , v ) for all ( u , v ) E .

Observe that

inf { R N | ϕ | p : ϕ C 0 ( R N , R ) , ϕ L α ( R N ) = 1 } = 0

and

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

Then, for any δ > 0 , there are ϕ δ , ψ δ C 0 ( R N , R ) with ϕ δ L α ( R N ) = ψ δ L β ( R N ) = 1 and supp ϕ δ , supp ψ δ B r δ ( 0 ) such that ϕ δ p p , ψ δ p p < δ .

Set e λ ( x ) = ( ϕ δ ( λ p x ) , ψ δ ( λ p x ) ) . Then supp e λ B λ 1 p r δ ( 0 ) . For t 0 , we get

Φ λ ( t e λ ) = t p p e λ E p a 0 λ t α R N | ϕ δ ( λ p x ) | α a 0 λ t β R N | ψ δ ( λ p x ) | β = λ 1 N p J λ ( t ϕ δ , t ψ δ ) ,

where

J λ ( u , v ) = 1 p R N ( | u | p + | v | p + ( A ( λ 1 p x ) + V ( λ 1 p x ) ) ( | u | p + | v | p ) ) a 0 R N ( | u | α + | v | β ) .

By direct computation, we have

max t 0 J λ ( t ϕ δ , t ψ δ ) α p p α ( α a 0 ) p α p { R N ( | ϕ δ | p + ( A ( λ 1 p x ) + V ( λ 1 p x ) ) | ϕ δ | p ) } α α p + β p p β ( β a 0 ) p β p { R N ( | ψ δ | p + ( A ( λ 1 p x ) + V ( λ 1 p x ) ) | ψ δ | p ) } β β p .

Since A ( 0 ) = 0 , V ( 0 ) = 0 and ϕ δ p p , ψ δ p p < δ , we know that there is Λ δ > 0 such that for all λ Λ δ , we have

max t 0 I λ ( t ϕ δ , t ψ δ ) ( α p p α ( α a 0 ) p α p ( 5 δ ) α α p + β p p β ( β a 0 ) p β p ( 5 δ ) β β p ) λ 1 N p . (4.1)

Lemma 4.3For any σ > 0 , there is Λ σ > 0 such that λ Λ σ , there is w ¯ λ E with w ¯ λ E > ρ λ , I λ ( w ¯ λ ) 0 and

max t 0 I λ ( t w ¯ λ ) σ λ 1 N p ,

where ρ λ is defined in Lemma 4.1.

Proof This proof is similar to that of Lemma 4.3 in [7], so we omit the details. □

5 Proof of Theorem 2

Proof By using Lemma 4.3, for any σ > 0 with 0 < σ < α 0 , we choose Λ σ > 0 and define the mini-max level

c λ = inf γ Γ λ max t [ 0 , 1 ] I λ ( γ ( t ) ) σ λ 1 N p for all  λ Λ σ ,

where Γ λ = { γ C ( [ 0 , 1 ] , E ) : γ ( 0 ) = 0 , γ ( 1 ) = w ¯ λ } .

By Proposition 3.1, we know that I λ satisfies the ( PS ) c λ condition. Hence, by the mountain-pass theorem, there is ( u λ , v λ ) E such that I λ ( u λ , v λ ) = c λ and I λ ( u λ , v λ ) = 0 . This shows ( u λ , v λ ) is a weak solution of (2.1).

Moreover, note that I λ ( u λ , v λ ) σ λ 1 N p and I λ ( u λ , v λ ) = 0 . Then

I λ ( u λ , v λ ) = I λ ( u λ , v λ ) 1 θ I λ ( u λ , v λ ) ( u λ , v λ ) = ( 1 p 1 θ ) ( u λ , v λ ) E p + ( 1 θ 1 p ) λ R N K ( x ) ( | u λ | p + | v λ | p ) + λ R N ( 1 θ ( | u λ | p H s ( | u λ | p , | v λ | p ) + | v λ | p H t ( | u λ | p , | v λ | p ) ) 1 p H ( | u λ | p , | v λ | p ) ) ( 1 p 1 θ ) ( u λ , v λ ) E p .

Furthermore, together with the diamagnetic inequality, we prove that ( u λ , v λ ) satisfies the estimate (2.2). The proof is complete. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally in this article. All authors 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 is supported by the National Natural Science Foundation of China (11271364) and the Fundamental Research Funds for the Central Universities (2012QNA46).

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