Open Access

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

Boundary Value Problems20132013:11

DOI: 10.1186/1687-2770-2013-11

Received: 5 October 2012

Accepted: 9 January 2013

Published: 24 January 2013

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, [115] 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 [1722].

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 [2327] 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 [2832]. 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 1 Suppose 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 2 Suppose 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 3.1 Let the assumptions of Theorem  2 be satisfied. There exists a constant α 0 0 independent of λ such 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 3.2 Assume 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 3.1 Let 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 space E.

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 3.2 Let 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 3.3 One 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 3.4 Let { ( 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
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equ11_HTML.gif
(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 4.1 Let 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 4.2 Under 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 4.3 For 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. □

Declarations

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).

Authors’ Affiliations

(1)
Department of Mathematics, China University of Mining and Technology

References

  1. Ambrosetti A, Rabinowitz PH: Dual variational methods in critical point theory and applications. J. Funct. Anal. 1973, 14: 349-381. 10.1016/0022-1236(73)90051-7MathSciNetView ArticleGoogle Scholar
  2. Benci V: On critical point theory of indefinite functions in the presence of symmetries. Trans. Am. Math. Soc. 1982, 274: 533-572. 10.1090/S0002-9947-1982-0675067-XMathSciNetView ArticleGoogle Scholar
  3. Brézis H, Nirenberg L: Positive solutions of nonlinear elliptic equation involving critical Sobolev exponents. Commun. Pure Appl. Math. 1983, 16: 437-477.View ArticleGoogle Scholar
  4. Cingolani S, Nolasco M: Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equation. Proc. R. Soc. Edinb. 1998, 128: 1249-1260. 10.1017/S030821050002730XMathSciNetView ArticleGoogle Scholar
  5. Del Pino M, Felmer PL: Semi-classical states for nonlinear Schrödinger equations. J. Funct. Anal. 1997, 149: 245-265. 10.1006/jfan.1996.3085MathSciNetView ArticleGoogle Scholar
  6. Del Pino M, Felmer PL: Multi-peak bound states for nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1998, 15: 127-149. 10.1016/S0294-1449(97)89296-7MathSciNetView ArticleGoogle Scholar
  7. Ding YH, Lin FH: Solutions of perturbed Schrödinger equations with critical nonlinearity. Calc. Var. 2007, 30: 231-249. 10.1007/s00526-007-0091-zMathSciNetView ArticleGoogle Scholar
  8. Guedda M, Veron L: Quasilinear elliptic equations involving critical Sobolev exponents. Nonlinear Anal. 1989, 12: 879-902.MathSciNetView ArticleGoogle Scholar
  9. Jeanjean L, Tanaka K: Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Calc. Var. Partial Differ. Equ. 2004, 21: 287-318.MathSciNetView ArticleGoogle Scholar
  10. Kang X, Wei JC: On interacting bumps of semi-classical states of nonlinear Schrödinger equations. Adv. Differ. Equ. 2000, 5: 899-928.MathSciNetGoogle Scholar
  11. Li YY: On a singularly perturbed elliptic equation. Adv. Differ. Equ. 1997, 2: 955-980.Google Scholar
  12. Oh YG: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential. Commun. Math. Phys. 1990, 131: 223-253. 10.1007/BF02161413View ArticleGoogle Scholar
  13. Oh YG:Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of the class ( V ) a . Commun. Partial Differ. Equ. 1988, 13: 1499-1519. 10.1080/03605308808820585View ArticleGoogle Scholar
  14. Pistoia A: Multi-peak solutions for a class of some results on a class of nonlinear Schrödinger equations. Nonlinear Differ. Equ. Appl. 2002, 9: 69-91. 10.1007/s00030-002-8119-8MathSciNetView ArticleGoogle Scholar
  15. Floer A, Weinstein A: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal. 1986, 69: 397-408. 10.1016/0022-1236(86)90096-0MathSciNetView ArticleGoogle Scholar
  16. Esteban M, Lions PL: Stationary solutions of nonlinear Schrödinger equation with an external magnetic field. In Partial Differential Equations and the Calculus of Variations, Essays in Honor of E. De Giorgi. Brikhäuser, Basel; 1990:369-408.Google Scholar
  17. Arioli G, Szulkin A: A semilinear Schrödinger equation in the presence of a magnetic field. Arch. Ration. Mech. Anal. 2003, 170: 277-295. 10.1007/s00205-003-0274-5MathSciNetView ArticleGoogle Scholar
  18. Bartsch T, Dancer EN, Peng S: On multi-bump semi-classical bound states of nonlinear Schrödinger equations with electromagnetic fields. Adv. Differ. Equ. 2006, 11: 781-812.MathSciNetGoogle Scholar
  19. Cao D, Tang Z: Existence and uniqueness of multi-bump bound states of nonlinear Schrödinger equations with electromagnetic fields. J. Differ. Equ. 2006, 222: 381-424. 10.1016/j.jde.2005.06.027MathSciNetView ArticleGoogle Scholar
  20. Cingolani S: Semiclassical stationary states of nonlinear Schrödinger equation with an external magnetic field. J. Differ. Equ. 2003, 188: 52-79. 10.1016/S0022-0396(02)00058-XMathSciNetView ArticleGoogle Scholar
  21. Han P: Solutions for singular critical growth Schrödinger equation with magnetic field. Port. Math. 2006, 63: 37-45.MathSciNetGoogle Scholar
  22. Kurata K: Existence and semi-classical limit of the least energy solution to a nonlinear Schrödinger equation with electromagnetic fields. Nonlinear Anal. 2000, 41: 763-778. 10.1016/S0362-546X(98)00308-3MathSciNetView ArticleGoogle Scholar
  23. Alves CO, Ding YH: Multiplicity of positive solutions to a p -Laplacian equation involving critical nonlinearity. J. Math. Anal. Appl. 2003, 279: 508-521. 10.1016/S0022-247X(03)00026-XMathSciNetView ArticleGoogle Scholar
  24. Gloss E: Existence and concentration of bound states for a p -Laplacian equation in R N . Adv. Nonlinear Stud. 2010, 10: 273-296.MathSciNetGoogle Scholar
  25. Liu CG, Zheng YQ: Existence of nontrivial solutions for p -Laplacian equations in R N . J. Math. Anal. Appl. 2011, 380: 669-679. 10.1016/j.jmaa.2011.02.064MathSciNetView ArticleGoogle Scholar
  26. Manásevich R, Mawhin J: Boundary value problems for nonlinear perturbations of vector p -Laplacian-like operators. J. Korean Math. Soc. 2000, 5: 665-685.Google Scholar
  27. Zhang HX, Liu WB: Existence of nontrivial solutions to perturbed p -Laplacian system in R N involving critical nonlinearity. Bound. Value Probl. 2012., 2012: Article ID 53Google Scholar
  28. Dinu TL: Entire solutions of Schrödinger systems with discontinuous nonlinearity and sign-changing potential. Math. Model. Anal. 2006, 13(3):229-242.MathSciNetGoogle Scholar
  29. Dinu TL: Entire solutions of multivalued nonlinear Schrödinger equations in Sobolev space with variable exponent. Nonlinear Anal. 2006, 65(7):1414-1424. 10.1016/j.na.2005.10.022MathSciNetView ArticleGoogle Scholar
  30. Gazzola F, Radulescu V: A nonsmooth critical point theory approach to some nonlinear elliptic equations in unbounded domains. Differ. Integral Equ. 2000, 13: 47-60.MathSciNetGoogle Scholar
  31. Ghanmi A, Maagli H, Radulescu V, Zeddini N: Large and bounded solutions for a class of nonlinear Schrödinger stationary systems. Anal. Appl. 2009, 7: 391-404. 10.1142/S0219530509001463MathSciNetView ArticleGoogle Scholar
  32. Rabinowitz PH: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys. 1992, 43: 270-291. 10.1007/BF00946631MathSciNetView ArticleGoogle Scholar
  33. Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems. Springer, New York; 1989.View ArticleGoogle Scholar
  34. Ghoussoub N, Yuan C: Multiple solutions for quasi-linear PDES involving the critical Sobolev and Hardy exponents. Trans. Am. Math. Soc. 2000, 352: 5703-5743. 10.1090/S0002-9947-00-02560-5MathSciNetView ArticleGoogle Scholar
  35. Li YY, Guo QQ, Niu PC: Global compactness results for quasilinear elliptic problems with combined critical Sobolev-Hardy terms. Nonlinear Anal. 2011, 74: 1445-1464. 10.1016/j.na.2010.10.018MathSciNetView ArticleGoogle Scholar
  36. Brézis H, Lieb E: A relation between pointwise convergence of functions and convergence of functional. Proc. Am. Math. Soc. 1983, 88: 486-490.View ArticleGoogle Scholar
  37. Pucci P, Radulescu V: The impact of the mountain pass theory in nonlinear analysis: a mathematical survey. Boll. Unione Mat. Ital., Ser. IX 2010, 3: 543-584.MathSciNetGoogle Scholar
  38. Radulescu V Contemporary Mathematics and Its Applications. In Qualitative Analysis of Nonlinear Elliptic Partial Differential Equations: Monotonicity, Analytic and Variational Methods. Hindawi Publ. Corp, New York; 2008.View ArticleGoogle Scholar

Copyright

© Zhang et al.; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. 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.