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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq1_HTML.gif 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equ1_HTML.gif
(1.1)

where Δ p , A u = div ( | u + i A ( x ) u | p 2 ( u + i A ( x ) u ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq2_HTML.gif, i is the imaginary unit, A ( x ) : R N R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq3_HTML.gif is real vector potential, 1 < p < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq4_HTML.gif, V ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq5_HTML.gif is a non-negative potential, p = N p / ( N p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq6_HTML.gif denotes the Sobolev critical exponent for N 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq7_HTML.gif and K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq8_HTML.gif is a bounded positive coefficient.

The scalar case corresponding to (1.1) has received considerable attention in recent years. For p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq9_HTML.gif and A ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq10_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equ2_HTML.gif
(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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equ3_HTML.gif
(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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equa_HTML.gif

Then ψ ( x , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq11_HTML.gif solves (1.3) if and only if u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq12_HTML.gif solves (1.2) with V ( x ) = W ( x ) E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq13_HTML.gif and ε 2 = ħ 2 2 m https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq14_HTML.gif.

The equation (1.2) has been extensively investigated in the literature based on various assumptions of the potential V ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq5_HTML.gif and the nonlinearity f ( x , u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq15_HTML.gif. See, for example, [115] and the references therein.

There are also many works dealing with the magnetic fields A ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq16_HTML.gif and p = 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq9_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq17_HTML.gif, most of the work, as we know, consider the scalar case which corresponds to (1.1) with A ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq10_HTML.gif. 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equ4_HTML.gif
(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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq18_HTML.gif, 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq19_HTML.gif 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq5_HTML.gif, A ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq18_HTML.gif, H ( s , t ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq20_HTML.gif and K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq8_HTML.gif throughout the paper:

( V 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq21_HTML.gif) V C ( R N , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq22_HTML.gif, V ( 0 ) = inf x R N V ( x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq23_HTML.gif and there exists b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq24_HTML.gif such that the set ν b : = { x R N : V ( x ) < b } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq25_HTML.gif has finite Lebesgue measure;

( A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq26_HTML.gif) A C ( R N , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq27_HTML.gif and A ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq28_HTML.gif;

( K 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq29_HTML.gif) K ( x ) C ( R N , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq30_HTML.gif, 0 < inf K sup K < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq31_HTML.gif;

( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq32_HTML.gif) H ( s , t ) C 1 ( R + × R + , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq33_HTML.gif and H s , H t = o ( | s | + | t | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq34_HTML.gif as | s | + | t | 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq35_HTML.gif;

( H 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq36_HTML.gif) there exist c 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq37_HTML.gif and p < α < p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq38_HTML.gif such that
| H s ( s , t ) | , | H t ( s , t ) | c 1 ( 1 + | s | α p p + | t | α p p ) ; https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equb_HTML.gif

( H 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq39_HTML.gif) there are a 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq40_HTML.gif, θ ( p , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq41_HTML.gif and α , β > p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq42_HTML.gif such that H ( s , t ) p a 0 ( | s | α p + | t | β p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq43_HTML.gif and 0 < θ p H ( s , t ) s H s + t H t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq44_HTML.gif.

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

Theorem 1 Suppose that the assumptions ( V 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq21_HTML.gif), ( A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq26_HTML.gif), ( K 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq29_HTML.gif) and ( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq32_HTML.gif)-( H 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq39_HTML.gif) hold. Then for any σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq45_HTML.gif, there is ε σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq46_HTML.gif such that if ε < ε σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq47_HTML.gif, the problem (1.1) has at least one solution ( u ε , v ε ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq48_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equc_HTML.gif
Setting λ = ε p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq49_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equ5_HTML.gif
(2.1)

We are going to prove the following result.

Theorem 2 Suppose that the assumptions ( V 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq21_HTML.gif), ( A 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq26_HTML.gif), ( K 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq29_HTML.gif) and ( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq32_HTML.gif)-( H 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq39_HTML.gif) hold. Then for any σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq45_HTML.gif, there is Λ σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq50_HTML.gif such that if λ > Λ σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq51_HTML.gif, the problem (2.1) has at least one solution ( u λ , v λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq52_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equ6_HTML.gif
(2.2)
For convenience, we quote the following notations. Let E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq53_HTML.gif denote the Banach space
E λ , A = { u W 1 , p ( R N ) : R N λ V ( x ) | u | p < } , λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equd_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Eque_HTML.gif

Set E = E λ , A × E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq54_HTML.gif and ( u , v ) E p = u λ , A p + v λ , A p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq55_HTML.gif for any ( u , v ) E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq56_HTML.gif.

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 | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equ7_HTML.gif
(2.3)

(the bar denotes complex conjugation). This inequality shows that if u E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq57_HTML.gif, then | u | W 1 , p ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq58_HTML.gif and therefore u L q ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq59_HTML.gif for any q [ p , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq60_HTML.gif. That is to say, if u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq61_HTML.gif in E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq53_HTML.gif, then u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq62_HTML.gif in L loc q ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq63_HTML.gif for any q [ p , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq60_HTML.gif and u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq62_HTML.gif a.e. in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq64_HTML.gif.

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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equf_HTML.gif

where G ( u , v ) = 1 p K ( x ) ( | u | p + | v | p ) + 1 p H ( | u | p , | v | p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq65_HTML.gif.

Under the assumptions of Theorem 2, standard arguments [33] show that I λ C 1 ( E λ , A , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq66_HTML.gif and its critical points are weak solutions of the equation (2.1).

3 ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq19_HTML.gif condition

We call a sequence { ( u n , v n ) } E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq67_HTML.gif a ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq19_HTML.gif sequence if I λ ( u n , v n ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq68_HTML.gif and I λ ( u n , v n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq69_HTML.gif strongly in E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq70_HTML.gif ( E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq70_HTML.gif is the dual space of E). I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq71_HTML.gif is said to satisfy the ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq19_HTML.gif condition if any ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq19_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq72_HTML.gif independent of λ such that, for any ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq19_HTML.gif sequence { ( u n , v n ) } E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq67_HTML.gif for I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq71_HTML.gif with ( u n , v n ) ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq73_HTML.gif, either ( u n , v n ) ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq74_HTML.gif or c I λ ( u , v ) α 0 λ 1 N p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq75_HTML.gif.

As a consequence, we obtain the following result.

Proposition 3.2 Assume that the assumptions of Proposition  3.1 hold, I λ ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq76_HTML.gif satisfies the ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq19_HTML.gif condition for all c α 0 λ 1 N p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq77_HTML.gif.

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 ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq78_HTML.gif is a ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq19_HTML.gif sequence of  I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq71_HTML.gif. Then c 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq79_HTML.gif and { ( u n , v n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq78_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equg_HTML.gif
Together with I λ ( u n , v n ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq68_HTML.gif and I λ ( u n , v n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq80_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq81_HTML.gif, we have
( 1 p 1 θ ) ( u n , v n ) E p c + o ( 1 ) + ε n ( u n , v n ) E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equh_HTML.gif

Then { ( u n , v n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq78_HTML.gif is bounded and c 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq79_HTML.gif. □

From Lemma 3.1, we may assume ( u n , v n ) ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq73_HTML.gif in E and ( u n , v n ) ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq74_HTML.gif in L loc q ( R N ) × L loc q ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq82_HTML.gif for any q [ p , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq60_HTML.gif and u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq62_HTML.gif, v n v https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq83_HTML.gif a.e. in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq64_HTML.gif.

Lemma 3.2 Let γ [ p , p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq84_HTML.gif. There is a subsequence { ( u n j , v n j ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq85_HTML.gif such that for any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq86_HTML.gif, there is r ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq87_HTML.gif with r r ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq88_HTML.gif
lim j sup B j B r ( | u n j | γ + | v n j | γ ) ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equi_HTML.gif

where B r : = { x R N : | x | r } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq89_HTML.gif.

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 + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq90_HTML.gif be a smooth function satisfying 0 η ( t ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq91_HTML.gif, η ( t ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq92_HTML.gif if t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq93_HTML.gif and η ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq94_HTML.gif if t 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq95_HTML.gif. Define u ˜ j ( x ) = η ( 2 | x | / j ) u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq96_HTML.gif, v ˜ j ( x ) = η ( 2 | x | / j ) v ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq97_HTML.gif. Obviously, we have
u u ˜ j E λ , A 0 and v v ˜ j E λ , A 0 as  j . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equ8_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equj_HTML.gif
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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equk_HTML.gif

uniformly in ( φ , ψ ) E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq98_HTML.gif with ( φ , ψ ) E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq99_HTML.gif.

Proof The local compactness of Sobolev embedding implies that for any r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq100_HTML.gif, 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equl_HTML.gif
uniformly in φ E λ , A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq101_HTML.gif. For any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq102_HTML.gif, there exists r ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq87_HTML.gif such that
lim j sup B j B r | u ˜ j | γ R N | u | γ ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equm_HTML.gif
for all r r ϵ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq88_HTML.gif. Together with the assumption ( H 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq36_HTML.gif) 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equn_HTML.gif
where c i https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq103_HTML.gif ( i = 1 , 2 , 3 , 4 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq104_HTML.gif) 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equo_HTML.gif

 □

Lemma 3.4 Let { ( u n , v n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq78_HTML.gif and { ( u ˜ n , v ˜ n ) } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq105_HTML.gif be as defined above. Then the following conclusions hold:
I λ ( u n u ˜ n , v n v ˜ n ) c I λ ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equp_HTML.gif
and
I λ ( u n u ˜ n , v n v ˜ n ) 0 in  E ( the dual space of  E ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equq_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equr_HTML.gif
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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equs_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equt_HTML.gif
Furthermore, using the fact I λ ( u n , v n ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq68_HTML.gif and I λ ( u ˜ n , v ˜ n ) I λ ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq106_HTML.gif, we obtain
I λ ( u n u ˜ n , v n v ˜ n ) c I λ ( u , v ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equu_HTML.gif
In order to prove I λ ( u n u ˜ n , v n v ˜ n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq107_HTML.gif in E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq108_HTML.gif, for any ( φ , ψ ) E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq98_HTML.gif, 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equv_HTML.gif
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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equw_HTML.gif
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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equx_HTML.gif
uniformly in ( φ , ψ ) E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq98_HTML.gif with ( φ , ψ ) E 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq99_HTML.gif. Together with Lemma 3.3, we have
I λ ( u n u ˜ n , v n v ˜ n ) 0 in  E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equy_HTML.gif

 □

Let u n 1 = u n u ˜ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq109_HTML.gif, v n 1 = v n v ˜ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq110_HTML.gif, then u n u = u n 1 + ( u ˜ n u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq111_HTML.gif, v n v = v n 1 + ( v ˜ n v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq112_HTML.gif. From (3.1), we get ( u n , v n ) ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq74_HTML.gif in E if and only if ( u n 1 , v n 1 ) ( 0 , 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq113_HTML.gif 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equz_HTML.gif
where K min = inf x R N K ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq114_HTML.gif. Furthermore, we get
( u n 1 , v n 1 ) p p N ( c I λ ( u , v ) ) λ K min + o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equ9_HTML.gif
(3.2)

Now, we consider the energy level of the functional I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq71_HTML.gif below which the ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq19_HTML.gif condition holds.

Let V b ( x ) : = max { V ( x ) , b } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq115_HTML.gif, where b is a positive constant in the assumption ( V 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq21_HTML.gif). Since the set ν b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq116_HTML.gif 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equ10_HTML.gif
(3.3)
In connection with the assumptions ( H 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq32_HTML.gif)-( H 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq39_HTML.gif) and the Young inequality, there exists C b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq117_HTML.gif 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equaa_HTML.gif
Proof of Proposition 3.1 Assume that ( u n , v n ) ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq118_HTML.gif, then
lim inf n ( u n 1 , v n 1 ) E > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equab_HTML.gif
and
c I λ ( u , v ) > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equac_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equad_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equae_HTML.gif
Set α 0 = S N p C b N p N 1 K min https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq119_HTML.gif, then
α 0 λ 1 N p c I λ ( u , v ) + o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equaf_HTML.gif

This completes the proof of Proposition 3.1. □

Proof of Proposition 3.2 Since c α 0 λ 1 N p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq77_HTML.gif, we have
c I λ ( u , v ) α 0 λ 1 N p I λ ( u , v ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equag_HTML.gif

In connection with I λ ( u , v ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq120_HTML.gif and Proposition 3.1, we complete this proof. □

4 The mountain-pass structure

In the following, we always consider λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq121_HTML.gif. We will prove that I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq71_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq122_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equah_HTML.gif
Proof By (3.4), for any δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq123_HTML.gif, there is C δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq124_HTML.gif such that
R N G ( u , v ) δ ( u p p + v p p ) + C δ ( u p p + v p p ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equai_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equaj_HTML.gif
In connection with u p p + v p p C 1 ( u , v ) E p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq125_HTML.gif, we may choose δ ( 2 p λ C 1 ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq126_HTML.gif such that
I λ ( u , v ) 1 2 p ( u , v ) E p λ C δ ( u p p + v p p ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equak_HTML.gif

The fact p > p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq127_HTML.gif implies the desired conclusion. □

Lemma 4.2 Under the assumptions of Lemma  4.1, for any finite dimensional subspace F E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq128_HTML.gif, we have
I λ ( u , v ) as  ( u , v ) F , ( u , v ) E . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equal_HTML.gif
Proof Together with the fact H ( s , t ) p a 0 ( | s | α p + | t | β p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq129_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equam_HTML.gif

Since all norms in a finite-dimensional space are equivalent and α , β > p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq130_HTML.gif, 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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equan_HTML.gif

Obviously, it follows that Φ λ C 1 ( E ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq131_HTML.gif and I λ ( u , v ) Φ λ ( u , v ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq132_HTML.gif for all ( u , v ) E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq56_HTML.gif.

Observe that
inf { R N | ϕ | p : ϕ C 0 ( R N , R ) , ϕ L α ( R N ) = 1 } = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equao_HTML.gif
and
inf { R N | ψ | p : ψ C 0 ( R N , R ) , ψ L β ( R N ) = 1 } = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equap_HTML.gif

Then, for any δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq123_HTML.gif, there are ϕ δ , ψ δ C 0 ( R N , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq133_HTML.gif with ϕ δ L α ( R N ) = ψ δ L β ( R N ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq134_HTML.gif and supp ϕ δ , supp ψ δ B r δ ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq135_HTML.gif such that ϕ δ p p , ψ δ p p < δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq136_HTML.gif.

Set e λ ( x ) = ( ϕ δ ( λ p x ) , ψ δ ( λ p x ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq137_HTML.gif. Then supp e λ B λ 1 p r δ ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq138_HTML.gif. For t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq139_HTML.gif, 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equaq_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equar_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equas_HTML.gif
Since A ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq28_HTML.gif, V ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq140_HTML.gif and ϕ δ p p , ψ δ p p < δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq141_HTML.gif, we know that there is Λ δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq142_HTML.gif such that for all λ Λ δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq143_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equ12_HTML.gif
(4.1)
Lemma 4.3 For any σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq144_HTML.gif, there is Λ σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq50_HTML.gif such that λ Λ σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq145_HTML.gif, there is w ¯ λ E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq146_HTML.gif with w ¯ λ E > ρ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq147_HTML.gif, I λ ( w ¯ λ ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq148_HTML.gif and
max t 0 I λ ( t w ¯ λ ) σ λ 1 N p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equat_HTML.gif

where ρ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq149_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq144_HTML.gif with 0 < σ < α 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq150_HTML.gif, we choose Λ σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq50_HTML.gif 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equau_HTML.gif

where Γ λ = { γ C ( [ 0 , 1 ] , E ) : γ ( 0 ) = 0 , γ ( 1 ) = w ¯ λ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq151_HTML.gif.

By Proposition 3.1, we know that I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq71_HTML.gif satisfies the ( PS ) c λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq152_HTML.gif condition. Hence, by the mountain-pass theorem, there is ( u λ , v λ ) E https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq153_HTML.gif such that I λ ( u λ , v λ ) = c λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq154_HTML.gif and I λ ( u λ , v λ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq155_HTML.gif. This shows ( u λ , v λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq156_HTML.gif is a weak solution of (2.1).

Moreover, note that I λ ( u λ , v λ ) σ λ 1 N p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq157_HTML.gif and I λ ( u λ , v λ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq155_HTML.gif. 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_Equav_HTML.gif

Furthermore, together with the diamagnetic inequality, we prove that ( u λ , v λ ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-11/MediaObjects/13661_2012_Article_268_IEq156_HTML.gif 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

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