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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq1_HTML.gif. 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equ1_HTML.gif
(1.1)

where Δ p , A u = div ( | Δ u + i A ( x ) u | p 2 ( u + i A ( x ) u ) ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq2_HTML.gif, i is the imaginary unit, A ( x ) : R N R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq3_HTML.gif is a real vector potential, 1 < p < N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq4_HTML.gif, p * = N p / ( N p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq5_HTML.gif denotes the Sobolev critical exponent and N 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq6_HTML.gif.

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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equ2_HTML.gif
(1.2)

where ħ is Planck's constant, i is the imaginary unit, 2 * = 2 N N - 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq7_HTML.gif ( N 3 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq8_HTML.gif) is the critical exponent, A ( x ) : R N R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq9_HTML.gif is a real vector potential, B = curl A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq10_HTML.gif and W ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq11_HTML.gif 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equa_HTML.gif
where ħ is a sufficiently small constant, E is a real number, and u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq12_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equ3_HTML.gif
(1.3)

We can conduct the transition from quantum mechanics to classical mechanics by letting ħ 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq13_HTML.gif. Thus, the existence of semiclassical solutions has agreat charm in physical interest.

Problem (1.3) with A ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq14_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq15_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq16_HTML.gif and special classes of magnetic fields. Cingolani [5] proved that the magnetic potential A ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq17_HTML.gif only contributes to the phase factor of the solitarysolutions for ħ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq16_HTML.gif sufficiently small. For more results, we refer the readerto [1921] and the references therein.

For general p 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq18_HTML.gif, most of the works studied the existence results toequation (1.1) with A ( x ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq14_HTML.gif. 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq1_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq1_HTML.gif and critical nonlinearity. At the same time, we mustconsider complex-valued functions for the appearance of electromagnetic potential A ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq17_HTML.gif. 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 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq19_HTML.gif, A ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq17_HTML.gif, f ( x , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq20_HTML.gif and K ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq21_HTML.gif throughout the paper:

(V0) V ϵ C ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq22_HTML.gif, V ( 0 ) = inf x ϵ R N V ( x ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq23_HTML.gif, and there exists b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq24_HTML.gif such that the set ν b : = { x ϵ R N : V ( x ) < b } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq25_HTML.gif has a finite Lebesgue measure;

(A0) A C ( R N , R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq26_HTML.gif and A ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq27_HTML.gif;

(K0) K ( x ) C ( R N , R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq28_HTML.gif, 0 < inf K sup K < https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq29_HTML.gif;

(H1) f C ( R N × R + , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq30_HTML.gif and f ( x , s ) = o ( | s | ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq31_HTML.gif uniformly in x as s 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq32_HTML.gif;

(H2) there are c 1 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq33_HTML.gif and p < α < p * https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq34_HTML.gif such that | f ( x , s ) | c 1 ( 1 + | s | α - p p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq35_HTML.gif for all ( x , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq36_HTML.gif;

(H3) there exist a 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq37_HTML.gif, q > p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq38_HTML.gif and θ ( p , p * ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq39_HTML.gif such that F ( x , s ) p a 0 | s | q p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq40_HTML.gif and θ p F ( x , s ) f ( x , s ) s https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq41_HTML.gif for all ( x , s ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq36_HTML.gif, where F ( x , s ) = ʃ 0 s f ( x , t ) d t https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq42_HTML.gif.

Our main result is the following.

Theorem 1 Assume that (V0), (A0), (K0)and (H1)-(H3) hold. Then forany σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq43_HTML.gif, there exists ε > σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq44_HTML.gifsuch that if ε ε σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq45_HTML.gif, equation (1.1) has at least one positiveleast energy solution u ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq46_HTML.gif, which satisfies
θ p p θ ʃ R N ( ε p | u ε | p + V ( x ) | u ε | p ) σ ε N . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equb_HTML.gif

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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq47_HTML.gif. 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equ4_HTML.gif
(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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq43_HTML.gif, there exists λ σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq48_HTML.gifsuch that if λ > λ σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq49_HTML.gif, then equation (2.1) has at least onesolution of least energy u ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq50_HTML.gifsatisfying
θ - p p θ ʃ R N ( | u λ | p + λ V ( x ) | u λ | p ) σ λ 1 - N p . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equ5_HTML.gif
(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 } , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equc_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equd_HTML.gif
It is known that E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq51_HTML.gif is the closure of C 0 ( R N , C ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq52_HTML.gif. Similar to the diamagnetic inequality [10], we have the following inequality
| | u ( x ) | | | u + i λ 1 p A u | . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Eque_HTML.gif
In fact, since A ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq17_HTML.gif 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 | https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equ6_HTML.gif
(2.3)

(the bar denotes a complex conjugation). This inequality implies that if u E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq53_HTML.gif, then | u | W 1 , p ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq54_HTML.gif, and, therefore, u L q ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq55_HTML.gif for any q [ p , p * ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq56_HTML.gif. That is, if u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq57_HTML.gif in E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq51_HTML.gif, then u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq58_HTML.gif in L loc q ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq59_HTML.gif for any q ϵ [ p , p * ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq60_HTML.gif and u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq58_HTML.gif a.e. in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq1_HTML.gif.

Solutions of (2.1) will be sought in the Sobolev space E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq51_HTML.gif 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equf_HTML.gif

where G ( x , u ) = 1 p * K ( x ) | u | p * + 1 p F ( x , | u | p ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq61_HTML.gif.

It is easy to see that I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq62_HTML.gif is a C 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq63_HTML.gif-functional on E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq51_HTML.gif[29].

3 Behavior of ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq64_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq64_HTML.gifsequence { u n } E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq65_HTML.giffor I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq62_HTML.gif, we get that c 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq66_HTML.gifand { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq67_HTML.gifis bounded in the space E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq51_HTML.gif.

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 ) ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equg_HTML.gif

In connection with the facts that I λ ( u n ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq68_HTML.gif and I ' λ ( u n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq69_HTML.gif as n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq70_HTML.gif, we obtain that the ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq64_HTML.gif sequence { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq67_HTML.gif is bounded in E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq51_HTML.gif, and the energy level c 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq66_HTML.gif.

Next, let { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq67_HTML.gif denote a ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq64_HTML.gif sequence. By Lemma 3.1, it is bounded, thus, without lossof generality, we may assume that u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq57_HTML.gif in E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq51_HTML.gif. Furthermore, passing to a subsequence, we have u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq58_HTML.gif in L loc q ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq59_HTML.gif for any q ϵ [ p , p * ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq56_HTML.gif and u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq58_HTML.gif a.e. in R N https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq1_HTML.gif.

Lemma 3.2 For any s ϵ [ p , p * ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq71_HTML.gif, there is a subsequence { u n j } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq72_HTML.gifsuch that for any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq73_HTML.gif, there exists r ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq74_HTML.gifwith
lim j sup ʃ B j \ B r | u n j | s ε for any r r ε , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equh_HTML.gif

where B r : = { x ϵ R N : | x | r } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq75_HTML.gif.

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

Let η C ( R + ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq76_HTML.gif be a smooth function satisfying 0 η ( t ) 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq77_HTML.gif, η ( t ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq78_HTML.gif if t 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq79_HTML.gif and η ( t ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq80_HTML.gif if t 2 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq81_HTML.gif. Define u ~ j ( x ) = η ( 2 | x | / j ) u ( x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq82_HTML.gif. It is not difficult to see that
ǁ u ǁ u ~ j ǁ λ , A 0 as j . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equi_HTML.gif
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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equj_HTML.gif

uniformly in ϕ E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq83_HTML.gifwith ǁ ϕ ǁ λ , A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq84_HTML.gif.

Proof By direct computation, we easily obtain u ~ j u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq85_HTML.gif in E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq51_HTML.gif. The local compactness of the Sobolev embedding impliesthat, for any r 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq86_HTML.gif, 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equk_HTML.gif
uniformly in ǁ ϕ ǁ λ , A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq84_HTML.gif. For any ε > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq73_HTML.gif, there is r ε 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq87_HTML.gif such that
lim j sup ʃ B j \ B r | u ~ j | s ʃ R N \ B r | u | s ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equl_HTML.gif
for all r r ε https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq88_HTML.gif. 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equm_HTML.gif

This proof is completed.

Lemma 3.4 One has along a subsequence
I λ ( u n - u ~ n ) c - I λ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equn_HTML.gif
and
I ' λ ( u n - u ~ n ) 0 in E λ - 1 ( the dual space of E λ ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equo_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equp_HTML.gif
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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equq_HTML.gif
and
lim n ʃ R N ( F ( x , | u n | p ) - F ( x , | u n - u ~ n | p ) - F ( x , | u ~ n | p ) ) = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equr_HTML.gif
We now observe that I λ ( u n ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq68_HTML.gif and I λ ( u ~ n ) I λ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq89_HTML.gif, which gives
I λ ( u n - u ~ n ) c - I λ ( u ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equs_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equt_HTML.gif
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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equu_HTML.gif

uniformly in ǁ ϕ ǁ λ , A 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq84_HTML.gif. Combining Lemma 3.3, we get I ' λ ( u n - u ~ n ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq90_HTML.gif. The proof is completed.

Let u n 1 = u n - u ~ n https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq91_HTML.gif, then u n - u = u n 1 + ( u ~ n - u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq92_HTML.gif. Therefore, u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq58_HTML.gif in E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq51_HTML.gif if and only if u n 1 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq93_HTML.gif in E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq51_HTML.gif.

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 * , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equv_HTML.gif
where K min = inf x ϵ R N K ( x ) > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq94_HTML.gif. Together with Lemma 3.4, one has
ǁ u n 1 ǁ p * p * N ( c - I λ ( u ) ) λ K min + o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equ7_HTML.gif
(3.1)

In the following, we consider the energy level of the functional I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq62_HTML.gif below which the ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq64_HTML.gif condition holds.

Denote V b ( x ) : = max { V ( x ) , b } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq95_HTML.gif, where b is the positive constant in assumption(V0). Since the set ν b https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq96_HTML.gif has a finite measure, combining the fact that u n 1 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq93_HTML.gif in L loc p ( R N ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq97_HTML.gif, we get
ʃ R N V ( x ) | u n 1 | p = ʃ R N V b ( x ) | u n 1 | p + o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equ8_HTML.gif
(3.2)
Furthermore, by (K0) and (H1)-(H3), there exists C b > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq98_HTML.gif 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 * . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equ9_HTML.gif
(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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equw_HTML.gif

Lemma 3.5 There exists α 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq99_HTML.gif (independent of λ) such that,for any ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq64_HTML.gifsequence { u n } E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq65_HTML.giffor I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq62_HTML.gifwith u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq57_HTML.gif, either u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq58_HTML.gifin E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq51_HTML.gifor c - I λ ( u ) α 0 λ 1 - N p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq100_HTML.gif.

Proof Arguing by contradiction, assume that u n u https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq101_HTML.gif, then
lim inf n ǁ u n 1 ǁ λ , A > 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equx_HTML.gif
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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equy_HTML.gif
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 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equz_HTML.gif
Denote α 0 = S N p C b - N p N 1 K min https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq102_HTML.gif, then
α 0 λ 1 - N p c - I λ ( u ) + o ( 1 ) . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equaa_HTML.gif

We obtain the desired conclusion.

Lemma 3.6 There exists a constant α 0 > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq99_HTML.gif (independent of λ) such that ifa ( PS ) c https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq64_HTML.gifsequence { u n } E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq65_HTML.giffor I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq62_HTML.gifsatisfies c α 0 λ 1 - N p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq103_HTML.gif, the sequence { u n } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq67_HTML.gifhas a strongly convergent subsequencein E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq51_HTML.gif.

Proof By the fact that I λ ( u ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq104_HTML.gif and Lemma 3.5, we easily get the requiredconclusion.

Now, we consider λ 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq105_HTML.gif and prove that the energy functional I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq62_HTML.gif possesses the mountain pass structure.

Lemma 3.7 Under the assumptions of Theorem 2, thereexist α λ , ρ λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq106_HTML.gifsuch 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equab_HTML.gif

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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq107_HTML.gif, we have
I λ ( u ) - , u ϵ F as ǁ u ǁ λ , A . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equac_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equad_HTML.gif

Since all norms in a finite-dimensional space are equivalent, in connection with q > p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq38_HTML.gif, we obtain the desired conclusion.

For λ large enough and c λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq108_HTML.gif small sufficiently, I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq62_HTML.gif satisfies ( PS ) c λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq109_HTML.gif 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equae_HTML.gif
It is easy to see that Ф λ ϵ C 1 ( E λ , A ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq110_HTML.gif and I λ ( u ) Ф λ ( u ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq111_HTML.gif for all u ϵ E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq53_HTML.gif. Note that
inf { ʃ R N | φ | p : φ ϵ C 0 ( R N , R ) , ǁ φ ǁ L q ( R N ) = 1 } = 0 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equaf_HTML.gif
For any δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq112_HTML.gif, there is φ δ ϵ C 0 ( R N , R ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq113_HTML.gif with ǁ φ δ ǁ L q ( R N ) = 1 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq114_HTML.gif and supp φ δ B r δ ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq115_HTML.gif such that ǁ φ δ ǁ p p < δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq116_HTML.gif. Let e λ ( x ) = φ δ ( λ p x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq117_HTML.gif, then supp e λ B λ - 1 p r δ ( 0 ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq118_HTML.gif. For any t 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq119_HTML.gif, 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 ) , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equag_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equah_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equai_HTML.gif
Observe that A ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq27_HTML.gif, V ( 0 ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq120_HTML.gif and ǁ φ δ ǁ p p < δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq116_HTML.gif. Therefore, there exists Ʌ δ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq121_HTML.gif such that for all λ Ʌ δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq122_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equ10_HTML.gif
(3.4)
Lemma 3.9 Under the assumptions of Theorem 2, forany σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq43_HTML.gif, there is Ʌ σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq123_HTML.gifsuch that for each λ Ʌ σ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq124_HTML.gif, there exists e - λ ϵ E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq125_HTML.gifwith ǁ e - λ ǁ λ , A > ρ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq126_HTML.gif, I λ ( e - λ ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq127_HTML.gifand
max t 0 I λ ( t e ¯ λ ) σ λ 1 - N p , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equaj_HTML.gif

where ρ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq128_HTML.gifis defined in Lemma 3.7.

Proof For any σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq43_HTML.gif, we can choose δ < 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq129_HTML.gif so small that
q - p p q ( q a 0 ) p q - p ( 5 δ ) q q - p σ . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equak_HTML.gif

Denote e λ ( x ) = φ δ ( λ p x ) https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq117_HTML.gif and Ʌ σ = Ʌ δ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq130_HTML.gif. Let t ¯ λ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq131_HTML.gif be such that t ¯ λ ǁ e λ ǁ λ , A > ρ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq132_HTML.gif and I λ ( t e λ ) 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq133_HTML.gif for all t t ¯ λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq134_HTML.gif. Then, combining (3.4), e ¯ λ = t ¯ λ e λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq135_HTML.gif 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 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq43_HTML.gif with 0 < σ < α 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq136_HTML.gif, we choose Ʌ σ > 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq123_HTML.gif 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 , https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equal_HTML.gif

where Г λ = { γ ϵ C ( [ 0 , 1 ] , E λ , A ) : γ ( 0 ) = 0 , γ ( 1 ) = e - λ } https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq137_HTML.gif.

Lemma 3.6 shows that I λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq62_HTML.gif satisfies ( PS ) c λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq109_HTML.gif condition. Therefore, by the mountain pass theorem, thereexists u λ ϵ E λ , A https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq138_HTML.gif, which satisfies I λ ( u λ ) = c λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq139_HTML.gif and I λ ( u λ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq140_HTML.gif. That is, u λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq50_HTML.gif is a weak solution of (2.1). Furthermore, it is well knownthat u λ https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq50_HTML.gif is the least energy solution of equation (2.1).

Moreover, together with I λ ( u λ ) σ λ 1 - N p https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq141_HTML.gif and I λ ( u λ ) = 0 https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_IEq140_HTML.gif, 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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equam_HTML.gif
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 . https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-217/MediaObjects/13661_2013_Article_533_Equan_HTML.gif

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

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© Zhang and Jiang; licensee Springer. 2013

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