Open Access

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

Boundary Value Problems20132013:217

DOI: 10.1186/1687-2770-2013-217

Received: 3 April 2013

Accepted: 28 August 2013

Published: 5 November 2013

Abstract

In this paper, we consider a perturbed p-Laplacian equation with criticalnonlinearity and magnetic fields on R N . By using the variational method, we establish theexistence 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 thefollowing perturbed p-Laplacian equation with critical nonlinearity and magnetic fields of the form
‒ε p Δ p , A u + V ( x ) | u | p 2 u = K ( x ) | u | p * 2 u + f ( x , | u | p ) | u | p 2 u , x ϵ R N ,
(1.1)

where Δ p , A u = div ( | Δ 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 Schrödinger equation withmagnetic fields of the form
i h δ ψ δ t = ħ 2 2 m ( Δ + i A ( x ) ) 2 ψ + W ( x ) ψ K ( x ) | ψ | 2 _ ψ 2 ψ - h ( x , | ψ | 2 ) ψ for x x ϵ R N ,
(1.2)

where ħ 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
ψ ( x , t ) = exp ( - i E t ħ ) u ( x ) ,
where ħ 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 ) + λ ( W ( x ) - E ) u ( x ) = λ K ( x ) | u | 2 * - 2 u + λ h ( x , | u | 2 ) u , x ϵ R N .
(1.3)

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

Problem (1.3) with A ( x ) 0 has an extensive literature. Different approaches havebeen taken to investigate this problem under various hypotheses on the potential andnonlinearity. See for example [118] and the references therein. The above-mentioned papers mostly concentrated onthe nonlinearities with subcritical conditions. Floer and Weinstein in [11] 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 [14] 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 [4]. Ding and Lin [8] 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 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 forfixed ħ > 0 and special classes of magnetic fields. Cingolani [5] proved that the magnetic potential A ( x ) only contributes to the phase factor of the solitarysolutions for ħ > 0 sufficiently small. For more results, we refer the readerto [1921] and the references therein.

For general p 1 , most of the works studied the existence results toequation (1.1) with A ( x ) 0 . See, for example, [2228] 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 R N 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 R N and critical nonlinearity. At the same time, we mustconsider complex-valued functions for the appearance of electromagnetic potential A ( x ) . To overcome this difficulty, we chiefly follow the ideasof [5]. 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 V ( x ) , A ( x ) , f ( x , s ) and K ( x ) throughout the paper:

(V0) V ϵ C ( R N ) , V ( 0 ) = inf x ϵ R N V ( x ) = 0 , and there exists b > 0 such that the set ν b : = { x ϵ 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 < α < p * such that | f ( x , s ) | c 1 ( 1 + | s | α - p p ) for all ( x , s ) ;

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

Our main result is the following.

Theorem 1 Assume that (V0), (A0), (K0)and (H1)-(H3) hold. Then forany σ > 0 , there exists ε > σ such that if ε ε σ , equation (1.1) has at least one positiveleast energy solution u ε , which satisfies
θ p p θ ʃ R N ( ε p | u ε | p + V ( x ) | u ε | p ) σ ε 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 thelast section.

2 Preliminaries

Let λ = ε - p . Equation (1.1) reads then as
- Δ p , A u + λ V ( x ) | u | p - 2 u = λ K ( x ) | u | p * - 2 u + λ f ( x , | u | p ) | u | p - 2 u , x ϵ R N .
(2.1)

We are going to prove the following result.

Theorem 2 Assume that (V0), (A0), (K0)and (H1)-(H3) are satisfied. Then forany σ > 0 , there exists λ σ > 0 such that if λ > λ σ , then equation (2.1) has at least onesolution of least energy u ε satisfying
θ - p p θ ʃ R N ( | u λ | p + λ V ( x ) | u λ | p ) σ λ 1 - N p .
(2.2)
In order to prove these theorems, we introduce the space
E λ , A = { u ϵ W 1 , p ( R N , C ) : ʃ R N λ V ( x ) | u | p < , λ > 0 } ,
equipped with the norm
|| u || λ , A = ( ʃ R N ( | u + i λ 1 p A ( x ) u | p + λ V ( x ) | u | p ) ) 1 p .
It is known that E λ , 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 λ 1 p A u | .
In fact, since A ( x ) is real-valued, one has
| | u ( x ) | | | u u ¯u | u | | = | Re ( u + i λ 1 p A u ) u u | u | | | u + i λ 1 p A u |
(2.3)

(the bar denotes a complex conjugation). This inequality implies 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, if u n u in E λ , A , then u n u in L loc q ( R N ) for any q ϵ [ p , p * ) and u n u a.e. in R N .

Solutions of (2.1) will be sought in the Sobolev space E λ , A as critical points of the functional
I λ ( u ) = 1 p ʃ R N ( | u + i λ 1 p A u | p + λ V ( x ) | u | p ) - λ p * ʃ R N K ( x ) | u | p * - λ p ʃ R N F ( x , | u | p ) = 1 p ǁ u ǁ λ , A - λ ʃ N 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 λ is a C 1 -functional on E λ , 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 theproof of Theorem 2.

Lemma 3.1 Let (V0), (A0), (K0)and (H1)-(H3) be satisfied. Forthe ( PS ) c sequence { u n } E λ , A for I λ , we get that c 0 and { u n } is bounded in the space E λ , A .

Proof Under assumptions (K0) and (H3), we have
I λ ( u n ) - 1 θ I ' λ ( u n ) u n = ( 1 p - 1 θ ) ǁ u n ǁ p λ A p + ( 1 θ - 1 p * ) λ ʃ R N K ( x ) | u n | p * + λ ʃ R N ( 1 θ f ( x , | u n | p ) | u n | p - 1 p F ( x , | u n | p ) ) .

In connection with the facts that I λ ( u n ) c and I ' λ ( u n ) 0 as n , we obtain that the ( PS ) c sequence { u n } is bounded in E λ , 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 lossof generality, we may assume that u n u in E λ , A . Furthermore, passing to a subsequence, we have u n u in L loc q ( R N ) for any q ϵ [ p , p * ) and u n u a.e. in R N .

Lemma 3.2 For any s ϵ [ p , p * ) , there is a subsequence { u n j } such that for any ε > 0 , there exists r ε > 0 with
lim j sup ʃ B j \ B r | u n j | s ε for any r r ε ,

where B r : = { x ϵ R N : | x | r } .

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

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 ) . It is not difficult to see that
ǁ u ǁ u ~ j ǁ λ , A 0 as j .
Lemma 3.3 One has
lim j sup Re ʃ 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 ) ϕ = 0

uniformly in ϕ E λ , A with ǁ ϕ ǁ λ , A 1 .

Proof By direct computation, we easily obtain u ~ j u in E λ , A . The local compactness of the Sobolev embedding impliesthat, for any r 0 , we have
lim j sup Re ʃ 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 ) - ϕ = 0
uniformly in ǁ ϕ ǁ λ , A 1 . For any ε > 0 , there is r ε 0 such that
lim j sup ʃ B j \ B r | u ~ j | s ʃ R N \ B r | u | s ε
for all r r ε . By the assumptions and the Hölder inequality, we have
lim j sup Re ʃ 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 ) ϕ = lim j sup Re ʃ 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 ) ϕ c 1 lim j sup ʃ B j \ B r ( | u n j | p - 1 + | u ~ j | p - 1 ) | ϕ | + c 2 lim j sup ʃ B j \ B r ( | u n j | ǁ ǁ 1 + | u ~ j | ǁ ǁ 1 ) | ϕ | c 1 lim j sup ( ǁ u n j ǁ L p p - 1 ( B j \ B r ) + ǁ u ~ j ǁ L p p - 1 ( B j \ B r ) ) ǁ ϕ ǁ L p ( B j \ B r ) + c 2 lim j sup ( ǁ u n j ǁ L α ( B j \ B r ) α - 1 + ǁ u ~ j ǁ L α ( B j \ B r ) α - 1 ) ǁ ϕ ǁ L α ( B j \ B r ) c 3 ε p - 1 p + c 4 ε α - 1 α .

This proof is completed.

Lemma 3.4 One has along a subsequence
I λ ( u n - u ~ n ) c - I λ ( u )
and
I ' λ ( u n - u ~ n ) 0 in E λ - 1 ( the dual space of E λ ) .
Proof Combining Lemma 2.1 of [30] and the arguments of [31], one has
I λ ( u n - u ~ n ) = I λ ( u n ) - I λ ( u ~ n ) + λ p * ʃ R N K ( x ) ( | u n | p * - | u n - u ~ n | p * - | u ~ n | p * ) + λ p * R N ( F ( x , | u n | p ) - F ( x , | u n - u ~ n | p ) - F ( x , | u ~ n | p ) ) + o ( 1 ) .
By the Brézis-Lieb lemma [32], we get
lim n ʃ R N K ( x ) ( | u n | p * - | u n - u ~ n | p * - | u ~ n | p * ) = 0
and
lim n ʃ 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 λ ( u n ) c and I λ ( u ~ n ) I λ ( u ) , which gives
I λ ( u n - u ~ n ) c - I λ ( u ) .
Moreover, by direct computation, we get
I λ ' ( u n - u ~ n ) ϕ = I ' λ ( u n ) ϕ - I ' λ ( u ~ n ) ϕ + λ Re ʃ 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 ʃ 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 ) ϕ + o ( 1 ) .
It then follows from the standard arguments that
lim n Re ʃ 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

uniformly in ǁ ϕ ǁ λ , A 1 . Combining Lemma 3.3, we get I ' λ ( 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 λ , A if and only if u n 1 0 in E λ , A .

Note that
I λ ( u n 1 ) - 1 p I λ ( u n 1 ) u n 1 = ( 1 p - 1 p * ) λ ʃ R N K ( x ) | u n 1 | p * + λ ʃ R N 1 p ( | u n 1 | p f ( x , | u n 1 | p ) - F ( x , | u n 1 | p ) ) λ N ʃ R N K ( x ) | u n 1 | p * λ N K min ǁ u n 1 ǁ p * p * ,
where K min = inf x ϵ R N K ( x ) > 0 . Together with Lemma 3.4, one has
ǁ u n 1 ǁ p * p * N ( c - I λ ( u ) ) λ K min + o ( 1 ) .
(3.1)

In the following, we consider the energy level of the functional I λ 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 ν b has a finite measure, combining the fact that u n 1 0 in L loc p ( R N ) , we get
ʃ R N V ( x ) | u n 1 | p = ʃ 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
ʃ R N ( K ( x ) | u n 1 | p * + | u n 1 | p f ( x , | u n 1 | p ) ) b ǁ u n 1 ǁ p p + C b ǁ u n 1 ǁ p * p * .
(3.3)
Let S be the best Sobolev constant of the immersion
S ǁ u ǁ p * p ʃ R N | u | p for all u ϵ W 1 , p ( R N ) .

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

Proof Arguing by contradiction, assume that u n u , then
lim inf n ǁ u n 1 ǁ λ , A > 0 .
Combining the Sobolev inequality, (3.2) and (3.3), we get
S ǁ u n 1 ǁ p * p ʃ R N | u n 1 | p ʃ R N ( | u n 1 + i λ 1 p A ( x ) u n 1 | p + λ V ( x ) | u n 1 | p ) - λ ʃ R N V ( x ) | u n 1 | p = λ ʃ R N ( K ( x ) | u n 1 | p * + | u n 1 | p f ( x , | u n 1 | p ) ) - λ ʃ R N V b ( x ) | u n 1 | p + o ( 1 ) λ b ǁ u n 1 ǁ p p + λ C b ǁ u n 1 ǁ p * p * - λ b ǁ u n 1 ǁ p p + o ( 1 ) = λ C b ǁ u n 1 ǁ p * p * + o ( 1 ) ,
which further gives
S λ C b ǁ u n 1 ǁ p * p * - p + o ( 1 ) λ C b ( N ( c - I λ ( u ) ) λ K min ) p N + o ( 1 ) = λ 1 - p N C b ( N K min ) p N ( c - I λ ( u ) ) p N + o ( 1 ) .
Denote α 0 = S N p C b - N p N 1 K min , then
α 0 λ 1 - N p c - I λ ( u ) + o ( 1 ) .

We obtain the desired conclusion.

Lemma 3.6 There exists a constant α 0 > 0 (independent of λ) such that ifa ( PS ) c sequence { u n } E λ , A for I λ satisfies c α 0 λ 1 - N p , the sequence { u n } has a strongly convergent subsequencein E λ , A .

Proof By the fact that I λ ( u ) 0 and Lemma 3.5, we easily get the requiredconclusion.

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

Lemma 3.7 Under the assumptions of Theorem 2, thereexist α λ , ρ λ > 0 such that
I λ ( u ) > 0 if 0 < ǁ u ǁ λ , A < ρ λ and I λ ( u ) α λ if ǁ u ǁ λ , A = ρ λ .

Proof The proof of Lemma 3.7 is similar to the one of Lemma 4.1 in [8].

Lemma 3.8 For any finite dimensional subspace F E λ , A , we have
I λ ( u ) - , u ϵ F as ǁ u ǁ λ , A .
Proof By assumptions (K0) and (H3), one has
I λ ( u ) 1 p ǁ u ǁ λ , A p - λ a 0 ǁ u ǁ q q for all u ϵ E λ , A .

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

For λ large enough and c λ small sufficiently, I λ satisfies ( PS ) c λ condition by Lemma 3.6. Furthermore, we will find specialfinite-dimensional subspace, by which we establish sufficiently small minimaxlevels.

Define the functional
Ф λ ( u ) = 1 p ʃ R N ( | u + i λ 1 p A ( x ) u | p + λ V ( x ) | u | p ) - λ a 0 ʃ R N | u | q .
It is easy to see that Ф λ ϵ C 1 ( E λ , A ) and I λ ( u ) Ф λ ( u ) for all u ϵ E λ , A . Note that
inf { ʃ R N | φ | p : φ ϵ C 0 ( R N , R ) , ǁ φ ǁ L q ( R N ) = 1 } = 0 .
For any δ > 0 , there is φ δ ϵ C 0 ( R N , R ) with ǁ φ δ ǁ L q ( R N ) = 1 and supp φ δ B r δ ( 0 ) such that ǁ φ δ ǁ p p < δ . Let e λ ( x ) = φ δ ( λ p x ) , then supp e λ B λ - 1 p r δ ( 0 ) . For any t 0 , we have
Ф λ ( t e λ ) = t p p ǁ e λ ǁ λ , A p - a 0 λ t q ʃ R N | φ δ ( λ p x ) | q = λ 1 - N p J λ ( t φ δ ) ,
where
J λ ( u ) = 1 p ʃ R N ( | u + i λ 1 p A ( x ) u | p + V ( λ - 1 p x ) | u | p ) a 0 ʃ R N | u | q .
We derive that
max t 0 J λ ( t φ δ ) q - p p q ( q a 0 ) p q - p ( ʃ R N ( | φ δ | p + ( A ( λ - 1 p x ) + V ( λ - 1 p x ) ) | φ δ | p ) ) q q - p .
Observe that A ( 0 ) = 0 , V ( 0 ) = 0 and ǁ φ δ ǁ p p < δ . Therefore, there exists Ʌ δ > 0 such that for all λ Ʌ δ , we have
max t 0 I λ ( t φ δ ) ( q - p p q ( q a 0 ) p q - p ( 5 δ ) q q - p ) λ 1 - N p .
(3.4)
Lemma 3.9 Under the assumptions of Theorem 2, forany σ > 0 , there is Ʌ σ > 0 such that for each λ Ʌ σ , there exists e - λ ϵ E λ , A with ǁ e - λ ǁ λ , A > ρ λ , I λ ( e - λ ) 0 and
max t 0 I λ ( t e ¯ λ ) σ λ 1 - N p ,

where ρ λ is defined in Lemma 3.7.

Proof For any σ > 0 , we can choose δ < 0 so small that
q - p p q ( q a 0 ) p q - p ( 5 δ ) q q - p σ .

Denote e λ ( x ) = φ δ ( λ p x ) and Ʌ σ = Ʌ δ . Let t ¯ λ > 0 be such that t ¯ λ ǁ e λ ǁ λ , A > ρ λ and I λ ( t e λ ) 0 for all t t ¯ λ . Then, combining (3.4), e ¯ λ = t ¯ λ e λ 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 σ > 0 with 0 < σ < α 0 , we choose Ʌ σ > 0 and define the minimax value
c λ = inf γ ϵ Г λ max t ϵ [ 0 , 1 ] I λ ( γ ( t ) ) with c λ σ λ 1 - N p for each λ Ʌ σ ,

where Г λ = { γ ϵ C ( [ 0 , 1 ] , E λ , A ) : γ ( 0 ) = 0 , γ ( 1 ) = e - λ } .

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

Moreover, together with I λ ( u λ ) σ λ 1 - N p and I λ ( u λ ) = 0 , we have
I λ ( u λ ) = I λ ( u λ ) - 1 θ I λ ( u λ ) ( u λ ) = ( 1 p - 1 θ ) ǁ u λ ǁ λ , A p + ( 1 θ - 1 p * ) λ ʃ R N K ( x ) | u λ | p * + λ ʃ R N ( 1 θ | u λ | p f ( x , | u λ | p ) - 1 p F ( x , | u λ | p ) ) ( 1 p - 1 θ ) ǁ u λ ǁ λ , A p .
By inequality (2.3), we obtain
θ - p p θ ʃ R N ( | u λ | p + λ V ( x ) | u λ | p ) σ λ 1 - N p .

The proof is complete.

Authors' contributions

The authors contributed equally in this article. They read and approved the finalmanuscript.

Declarations

Acknowledgements

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).

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 ArticleMATH
  2. Bartsch T, Dancer E, Peng S: On multi-bump semi-classical bound states of nonlinear Schrödinger equations withelectromagnetic fields. Adv. Differ. Equ. 2006, 11: 781-812.MathSciNetMATH
  3. 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 ArticleMATH
  4. Brézis H, Nirenberg L: Positive solutions of nonlinear elliptic equation involving critical Sobolevexponents. Commun. Pure Appl. Math. 1983, 16: 437-477.View ArticleMATH
  5. Cingolani S: Semiclassical stationary states of nonlinear Schrödinger equation with an externalmagnetic field. J. Differ. Equ. 2003, 188: 52-79. 10.1016/S0022-0396(02)00058-XMathSciNetView ArticleMATH
  6. Clapp M, Ding YH: Minimal nodal solutions of a Schrödinger equation with critical nonlinearity andsymmetric potential. Differ. Integral Equ. 2003, 16: 981-992.MathSciNetMATH
  7. Del Pino M, Felmer PL: Semi-classical states for nonlinear Schrödinger equations. J. Funct. Anal. 1997, 149: 245-265. 10.1006/jfan.1996.3085MathSciNetView ArticleMATH
  8. Ding YH, Lin FH: Solutions of perturbed Schrödinger equations with critical nonlinearity. Calc. Var. 2007, 30: 231-249. 10.1007/s00526-007-0091-zMathSciNetView ArticleMATH
  9. Ding YH, Szulkin A: Bound states for semilinear Schrödinger equations with sign-changing potential. Calc. Var. Partial Differ. Equ. 2007, 29: 397-419. 10.1007/s00526-006-0071-8MathSciNetView ArticleMATH
  10. Esteban M, Lions PL: Stationary solutions of nonlinear Schrödinger equation with an external magneticfield. In PDE and Calculus of Variations: In Honor of E De Giorgi. Birkhöuser, Basel; 1990:369-408.
  11. Floer A, Weinstein A: Nonspreading wave packets for the cubic Schrödinger equation with a boundedpotential. J. Funct. Anal. 1986, 69: 397-408. 10.1016/0022-1236(86)90096-0MathSciNetView ArticleMATH
  12. Grossi M: Some results on a class of nonlinear Schrödinger equations. Math. Z. 2000, 235: 687-705. 10.1007/s002090000158MathSciNetView ArticleMATH
  13. Jeanjean L, Tanaka K: Singularly perturbed elliptic problems with superlinear or asymptotically linearnonlinearities. Calc. Var. Partial Differ. Equ. 2004, 21: 287-318.MathSciNetView ArticleMATH
  14. Kang X, Wei J: On interacting bumps of semi-classical states of nonlinear Schrödingerequations. Adv. Differ. Equ. 2000, 5: 899-928.MathSciNetMATH
  15. Li YY: On a singularly perturbed elliptic equation. Adv. Differ. Equ. 1997, 2: 955-980.MathSciNetMATH
  16. Oh YG:Existence of semiclassical bound states of nonlinear Schrödinger equations withpotentials of the class ( V ) a . Commun. Partial Differ. Equ. 1988, 13: 1499-1519. 10.1080/03605308808820585View ArticleMATH
  17. Oh YG: On positive multi-lump bound states of nonlinear Schrödinger equations undermultiple well potential. Commun. Math. Phys. 1990, 131: 223-253. 10.1007/BF02161413View ArticleMATH
  18. Pistoia A: Multi-peak solutions for a class of some results on a class of nonlinearSchrödinger equations. Nonlinear Differ. Equ. Appl. 2002, 9: 69-91. 10.1007/s00030-002-8119-8MathSciNetView ArticleMATH
  19. Tang Z: On the least energy solutions of nonlinear Schrödinger equations withelectromagnetic fields. Comput. Math. Appl. 2007, 54: 627-637. 10.1016/j.camwa.2006.12.031MathSciNetView ArticleMATH
  20. Tang Z: Multi-bump bound states of nonlinear Schrödinger equations with electromagneticfields and critical frequency. J. Differ. Equ. 2008, 245: 2723-2748. 10.1016/j.jde.2008.07.035View ArticleMATH
  21. Wang F: On an electromagnetic Schrödinger equation with critical growth. Nonlinear Anal. 2008, 69: 4088-4098. 10.1016/j.na.2007.10.039MathSciNetView ArticleMATH
  22. Alves CO, Ding YH: Multiplicity of positive solutions to a p -Laplacian equation involvingcritical nonlinearity. J. Math. Anal. Appl. 2003, 279: 508-521. 10.1016/S0022-247X(03)00026-XMathSciNetView ArticleMATH
  23. El Khalil A, El Manouni S, Ouanan M: On some nonlinear elliptic problems for p -Laplacian in R N . NoDEA Nonlinear Differ. Equ. Appl. 2008, 15: 295-307. 10.1007/s00030-008-7027-yMathSciNetView ArticleMATH
  24. Fan XL: p ( x ) -Laplacian equations in R N with periodic data and nonperiodic perturbations. J. Math. Anal. Appl. 2008, 341: 103-119. 10.1016/j.jmaa.2007.10.006MathSciNetView ArticleMATH
  25. Habib SE, Tsouli N: On the spectrum of the p -Laplacian operator for Neumann eigenvalueproblems with weights. Electron. J. Differ. Equ. Conf. 2005, 14: 181-190.MATH
  26. Lê A: Eigenvalue problems for the p -Laplacian. Nonlinear Anal. 2006, 64: 1057-1099. 10.1016/j.na.2005.05.056MathSciNetView ArticleMATH
  27. 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 ArticleMATH
  28. Manásevich R, Mawhin J: Boundary value problems for nonlinear perturbations of vector p -Laplacian-like operators. J. Korean Math. Soc. 2000, 5: 665-685.MATH
  29. Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems. Springer, New York; 1989.View ArticleMATH
  30. Li YY, Guo QQ, Niu PC: Global compactness results for quasilinear elliptic problems with combinedcritical Sobolev-Hardy terms. Nonlinear Anal. 2011, 74: 1445-1464. 10.1016/j.na.2010.10.018MathSciNetView ArticleMATH
  31. Ghoussoub N, Yuan C: Multiple solutions for quasi-linear PDES involving the critical Sobolev and Hardyexponents. Trans. Am. Math. Soc. 2000, 352: 5703-5743. 10.1090/S0002-9947-00-02560-5MathSciNetView ArticleMATH
  32. Brézis H, Lieb E: A relation between pointwise convergence of functions and convergence offunctional. Proc. Am. Math. Soc. 1983, 88: 486-490. 10.1090/S0002-9939-1983-0699419-3View ArticleMATH

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