Least energy solutions for a quasilinear Schrödinger equation with potential well

Boundary Value Problems20132013:9

DOI: 10.1186/1687-2770-2013-9

Received: 23 October 2012

Accepted: 5 January 2013

Published: 21 January 2013

Abstract

In this paper, we consider the existence of least energy solutions for the following quasilinear Schrödinger equation:

http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equa_HTML.gif

with a ( x ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq1_HTML.gif having a potential well, where N 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq2_HTML.gif and λ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq3_HTML.gif is a parameter. Under suitable hypotheses, we obtain the existence of a least energy solution u λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq4_HTML.gif of ( E λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq5_HTML.gif) which localizes near the potential well int a 1 ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq6_HTML.gif for λ large enough by using the variational method and the concentration compactness method in an Orlicz space.

MSC:35J60, 35B33.

Keywords

quasilinear Schrödinger equation least energy solution Orlicz space concentration compactness method variational method

1 Introduction

Let us consider the following quasilinear Schrödinger equation:
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equb_HTML.gif

for sufficiently large λ, where N 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq2_HTML.gif.

Our assumptions on a ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq7_HTML.gif are as follows:

( a 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq8_HTML.gif) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq9_HTML.gif , the potential well Ω : = int a 1 ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq10_HTML.gif is a non-empty set and Ω ¯ = a 1 ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq11_HTML.gif;

( a 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq12_HTML.gif) There exists a constant M 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq13_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq14_HTML.gif , where μ denotes the Lebesgue measure on http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq15_HTML.gif .

Condition ( a 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq12_HTML.gif) is very weak in dealing with the operator Δ + ( λ a ( x ) + 1 ) I http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq16_HTML.gif on http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq15_HTML.gif , which was firstly used by Bartsch and Wang [1] in dealing with the semilinear Schrödinger equation.

Remark 1.1 Ω : = int a 1 ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq17_HTML.gif can be unbounded.

For f ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq18_HTML.gif, we assume that f is continuous and satisfies the following conditions:

( f 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq19_HTML.gif) lim u 0 + f ( u ) u = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq20_HTML.gif;

( f 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq21_HTML.gif) 0 f ( u ) C ( 1 + u p ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq22_HTML.gif for u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq23_HTML.gif, where C > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq24_HTML.gif is a constant and 4 < p + 1 < 2 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq25_HTML.gif, where 2 = 2 N N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq26_HTML.gif;

( f 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq27_HTML.gif) There is a number 4 < θ p + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq28_HTML.gif such that for all u > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq29_HTML.gif, we have f ( u ) u θ F ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq30_HTML.gif, where F ( u ) = 0 u f ( t ) d t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq31_HTML.gif.

Hypotheses ( a 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq8_HTML.gif), ( a 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq12_HTML.gif) and ( f 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq19_HTML.gif), ( f 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq21_HTML.gif), ( f 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq27_HTML.gif) will be maintained throughout this paper.

Solutions of ( E λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq5_HTML.gif) are related to the existence of the standing wave solutions of the following quasilinear Schrödinger equation:
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ1_HTML.gif
(1.1)

where V ( x ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq32_HTML.gif is a given potential, k is a real constant and f, h are real functions. We would like to mention that (1.1) appears more naturally in mathematical physics and has been derived as models of several physical phenomena corresponding to various types of h. For instance, the case h ( s ) = s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq33_HTML.gif was used for the superfluid film equation in plasma physics by Kurihara [2] (see also [3]); in the case of h ( s ) = ( 1 + s ) 1 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq34_HTML.gif, (1.1) was used as a model of the self-changing of a high-power ultrashort laser in matter (see [47] and references therein).

In recent years, much attention has been devoted to the quasilinear Schrödinger equation of the following form:
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ2_HTML.gif
(1.2)
For example, by using a constrained minimization argument, the existence of positive ground state solution was proved by Poppenberg, Schmitt and Wang [8]. Using a change of variables, Liu, Wang and Wang [9] used an Orlicz space to prove the existence of soliton solution of (1.2) via the mountain pass theorem. Colin and Jeanjean [10] also made use of a change of variables but worked in the Sobolev space http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq35_HTML.gif , they proved the existence of a positive solution for (1.2) from the classical results given by Berestycki and Lions [11]. By using the Nehari manifold method and the concentration compactness principle (see [12]) in the Orlicz space, Guo and Tang [13] considered the following equation:
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ3_HTML.gif
(1.3)

with a ( x ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq1_HTML.gif having a potential well and 4 < p + 1 < 2 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq25_HTML.gif, where 2 = 2 N N 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq26_HTML.gif is the critical Sobolev exponent, and they proved the existence of a ground state solution of (1.3) which localizes near the potential well int a 1 ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq36_HTML.gif for λ large enough. In [14], Guo and Tang also considered ground state solutions of the corresponding quasilinear Schrödinger systems for (1.3) by the same methods and obtained similar results. For the stability and instability results for the special case of (1.2), one can also see the paper by Colin, Jeanjean and Squassina [15].

It is worth pointing out that the existence of one-bump or multi-bump bound state solutions for the related semilinear Schrödinger equation (1.3) for k = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq37_HTML.gif has been extensively studied. One can see Bartsch and Wang [1], Ambrosetti, Badiale and Cingolani [16], Ambrosetti, Malchiodi and Secchi [17], Byeon and Wang [18], Cingolani and Lazzo [19], Cingolani and Nolasco [20], Del Pino and Felmer [21, 22], Floer and Weinstein [23], Oh [24, 25] and the references therein.

In this paper, based on the idea from Liu, Wang and Wang [9], we consider the more general equation ( E λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq5_HTML.gif), the existence of least energy solutions for equation ( E λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq5_HTML.gif) with a potential well int a 1 ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq36_HTML.gif for λ large is proved under the conditions ( a 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq8_HTML.gif), ( a 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq12_HTML.gif) and ( f 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq19_HTML.gif), ( f 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq21_HTML.gif), ( f 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq27_HTML.gif).

The paper is organized as follows. In Section 2, we describe our main result (Theorem 2.1). In Section 3, we give some preliminaries that will be used for the proof of the main result. Finally, Theorem 2.1 will be proved in Section 4.

Throughout this paper, we use the same C to denote different universal constants.

2 Main result

Let V λ ( x ) = λ a ( x ) + 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq38_HTML.gif. Formally, we define the following functional:
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ4_HTML.gif
(2.1)

for http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq39_HTML.gif . Note that under our assumptions, the functional J λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq40_HTML.gif is not well defined on X.

We follow the idea of [9] and make the following change of variable.

Let v = h ( u ) = 1 2 u 1 + u 2 + 1 2 ln ( u + 1 + u 2 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq41_HTML.gif, then d v = 1 + u 2 d u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq42_HTML.gif. Moreover, h ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq43_HTML.gif satisfies
h ( u ) { u , | u | 1 , 1 2 u | u | , | u | 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equc_HTML.gif
Since h ( u ) = 1 + u 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq44_HTML.gif, h ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq43_HTML.gif is strictly monotone and hence has an inverse function denoted by u = g ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq45_HTML.gif. Obviously,
g ( v ) { v , | v | 1 , 2 | v | v , | v | 1 , g ( v ) = 1 1 + g 2 ( v ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equd_HTML.gif
Let G ( v ) = g 2 ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq46_HTML.gif. Then it holds that
G ( v ) = g 2 ( v ) { v 2 , | v | 1 , 2 | v | , | v | 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Eque_HTML.gif
and G ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq47_HTML.gif is convex. Moreover, there exists C 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq48_HTML.gif such that G ( 2 v ) C 0 G ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq49_HTML.gif,
G ( v ) = 2 g ( v ) 1 + g 2 ( v ) , G ( v ) = 2 ( 1 + g 2 ( v ) ) 2 > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equf_HTML.gif
Now we introduce the Orlicz space (see [26])
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equg_HTML.gif
equipped with the norm
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equh_HTML.gif

Then E G λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq50_HTML.gif is a Banach space.

Let
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equi_HTML.gif
equipped with the norm
v λ = v L 2 + | v | G λ . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equj_HTML.gif
Using the change of variable, we define the functional Φ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq51_HTML.gif on H G λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq52_HTML.gif by
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ5_HTML.gif
(2.2)

where v + = max { v , 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq53_HTML.gif is the positive part of v.

Let
N λ : = { v H G λ { 0 } | Φ λ ( v ) , v = 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equk_HTML.gif
be the Nehari manifold and let
c λ : = inf v N λ Φ λ ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equl_HTML.gif

be the infimum of Φ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq51_HTML.gif on the Nehari manifold N λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq54_HTML.gif, where Φ λ ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq55_HTML.gif is the Gateaux derivative (see Proposition 3.3).

We say that u λ = g ( v λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq56_HTML.gif is a least energy solution of ( E λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq5_HTML.gif) if v λ N λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq57_HTML.gif such that c λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq58_HTML.gif is achieved.

Note that under our assumptions, for λ large enough, the following Dirichlet problem is a kind of a ‘limit’ problem:
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equm_HTML.gif

where Ω : = int a 1 ( 0 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq59_HTML.gif.

Similar to the definition of the least energy solution of ( E λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq5_HTML.gif), we can define the least energy solution of (D) which will be given in Section 4.

Our main result is as follows.

Theorem 2.1 Assume that ( a 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq8_HTML.gif), ( a 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq12_HTML.gif) and ( f 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq19_HTML.gif), ( f 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq21_HTML.gif), ( f 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq27_HTML.gif) are satisfied. Then for λ large, c λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq58_HTML.gif is achieved by a critical point v λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq60_HTML.gif of Φ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq51_HTML.gif such that u λ = g ( v λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq56_HTML.gif is a least energy solution of ( E λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq5_HTML.gif). Furthermore, for any sequence λ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq61_HTML.gif, { v λ n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq62_HTML.gif has a subsequence converging to v such that u = g ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq45_HTML.gif is a least energy solution of (D).

3 Preliminaries

In order to obtain the compactness of the functional Φ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq51_HTML.gif, we recall the following Lemmas 3.1 and 3.2 which can be found in [13].

Lemma 3.1 There exist two constants C 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq63_HTML.gif, C 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq64_HTML.gif such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ6_HTML.gif
(3.1)

for any v H G λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq65_HTML.gif.

Lemma 3.2 The map: v g ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq66_HTML.gif from H G λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq52_HTML.gif into http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq67_HTML.gif is continuous for 2 q 2 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq68_HTML.gif.

Now we consider the functional Φ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq51_HTML.gif defined on H G λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq52_HTML.gif by (2.2), the following Proposition 3.3 is due to [9].

Proposition 3.3
  1. (i)

    Φ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq51_HTML.gif is well defined on H G λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq52_HTML.gif;

     
  2. (ii)

    Φ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq51_HTML.gif is continuous in H G λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq52_HTML.gif;

     
  3. (iii)
    Φ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq51_HTML.gif is Gateaux differentiable, the Gateaux derivative Φ λ ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq55_HTML.gif for v H G λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq65_HTML.gif is a linear functional and Φ λ ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq55_HTML.gif is continuous in v in the strong-weak topology, that is, if v n v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq69_HTML.gif strongly in H G λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq52_HTML.gif, then Φ λ ( v n ) Φ λ ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq70_HTML.gif weakly. Moreover, the Gateaux derivative Φ λ ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq55_HTML.gif has the form
    http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ7_HTML.gif
    (3.2)
     

Recall that { v n } H G λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq71_HTML.gif is called a Palais-Smale sequence ((PS) c sequence in short) for Φ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq51_HTML.gif if Φ λ ( v n ) c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq72_HTML.gif and Φ λ ( v n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq73_HTML.gif in ( H G λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq74_HTML.gif, the dual space of H G λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq52_HTML.gif. We say that the functional Φ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq51_HTML.gif satisfies the (PS) c condition if any of (PS) c sequence (up to a subsequence, if necessary) { v n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq75_HTML.gif converges strongly in H G λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq52_HTML.gif.

Lemma 3.4 Any of (PS) c sequence { v n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq75_HTML.gif for Φ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq51_HTML.gif is bounded.

Proof Suppose that { v n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq75_HTML.gif is a (PS) c sequence of Φ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq51_HTML.gif. We have http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq76_HTML.gif and Φ λ ( v n ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq73_HTML.gif in the space ( H G λ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq74_HTML.gif.

Taking w n = g ( v n ) g ( v n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq77_HTML.gif, then | w n | = ( 1 + g 2 ( v n ) 1 + g 2 ( v n ) ) | v n | 2 | v n | http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq78_HTML.gif, we have w n λ C v n λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq79_HTML.gif, thus
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ8_HTML.gif
(3.3)
and
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ9_HTML.gif
(3.4)
Taking (3.4) 1 θ (3.3) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq80_HTML.gif yields
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equn_HTML.gif
Note that
1 2 1 θ ( 1 + g 2 ( v n ) 1 + g 2 ( v n ) ) > 1 2 2 θ = θ 4 2 θ , 1 2 1 θ = θ 2 2 θ , f ( u ) u > θ F ( u ) , θ > 4 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equo_HTML.gif
we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equp_HTML.gif
It follows from Lemma 3.1 that
C 1 min { v λ , v λ 2 } 2 θ θ 4 ( c + o ( 1 ) + o ( v n λ ) ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ10_HTML.gif
(3.5)

thus { v n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq75_HTML.gif is bounded in H G λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq52_HTML.gif.

Let K λ = { v H G λ | Φ λ ( v ) = 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq81_HTML.gif be the critical set of Φ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq51_HTML.gif. Suppose v K λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq82_HTML.gif, then it is easy to check that either v > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq83_HTML.gif or v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq84_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq15_HTML.gif by the definition of Φ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq51_HTML.gif and the strong maximum principle. □

Lemma 3.5 There exists 0 < σ < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq85_HTML.gif which is independent of λ such that v λ v 1 > σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq86_HTML.gif for all v K λ { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq87_HTML.gif and λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq88_HTML.gif.

Proof Assume that v λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq89_HTML.gif for any v K λ { 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq87_HTML.gif (otherwise, the conclusion is true). From ( f 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq19_HTML.gif), ( f 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq21_HTML.gif), we see that for any ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq90_HTML.gif, there is a constant C ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq91_HTML.gif such that f ( x , u ) ε u + C ε u p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq92_HTML.gif for u > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq29_HTML.gif. We have
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equq_HTML.gif

and we can easily deduce the desired result. □

Lemma 3.6 There exists a positive constant c 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq93_HTML.gif such that
lim sup n v n λ max { 2 θ ( θ 4 ) C 1 c , 2 θ ( θ 4 ) C 1 c } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equr_HTML.gif

and either c c 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq94_HTML.gif or c = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq95_HTML.gif if { v n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq75_HTML.gif is a (PS) c sequence for Φ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq51_HTML.gif, where C 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq96_HTML.gif is the constant in Lemma  3.1.

Proof Since { v n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq75_HTML.gif is a (PS) c sequence, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equs_HTML.gif
It follows from (3.5) that
lim sup n v n λ max { 2 θ ( θ 4 ) C 1 c , 2 θ ( θ 4 ) C 1 c } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equt_HTML.gif
On the other hand, for v n λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq97_HTML.gif, we have
o ( v n λ ) = Φ λ ( v n ) , g ( v n ) g ( v n ) C 1 min { v n λ , v n λ 2 } C 2 ( max { v n λ , v n λ 2 } ) p + 1 2 = C 1 v n λ 2 C 2 v n λ p + 1 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ11_HTML.gif
(3.6)
Thus, there exists σ 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq98_HTML.gif ( σ 1 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq99_HTML.gif) such that
Φ λ ( v n ) , g ( v n ) g ( v n ) 1 4 C 1 v n λ 2 for  v n λ σ 1 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ12_HTML.gif
(3.7)
Taking c 0 = σ 1 max { 2 θ ( θ 4 ) C 1 , 2 θ ( θ 4 ) C 1 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq100_HTML.gif, then we have
max { 2 θ ( θ 4 ) C 1 c , 2 θ ( θ 4 ) C 1 c } < σ 1 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equu_HTML.gif
if c < c 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq101_HTML.gif. It follows from (3.6) and (3.7) that
1 4 v n λ 2 o ( v n λ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equv_HTML.gif

hence, v n λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq102_HTML.gif and c = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq95_HTML.gif. Therefore, we have proved that there exists a constant c 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq103_HTML.gif such that either c c 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq94_HTML.gif or c = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq95_HTML.gif. □

Proposition 3.7 Let M > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq104_HTML.gif be a constant. Then for any ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq90_HTML.gif, there exist Λ ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq105_HTML.gif, R ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq106_HTML.gif such that
lim sup n B R ε c ( 1 2 f ( g ( ( v n ) + ) ) g ( ( v n ) + ) F ( g ( ( v n ) + ) ) ) d x ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equw_HTML.gif

if { v n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq75_HTML.gif is a (PS) c sequence of Φ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq51_HTML.gif with λ > Λ ε http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq107_HTML.gif, c M http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq108_HTML.gif, where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq109_HTML.gif .

Proof For all R > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq110_HTML.gif, let
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equx_HTML.gif
We have
A ( R ) g 2 ( v n ) d x 1 λ M 0 + 1 A ( R ) ( λ a ( x ) + 1 ) g 2 ( v n ) d x 1 λ M 0 + 1 A ( R ) ( | v n | 2 + ( λ a ( x ) + 1 ) g 2 ( v n ) ) d x 1 λ M 0 + 1 ( 2 θ θ 4 c + o ( v n λ ) ) = 1 λ M 0 + 1 ( 2 θ θ 4 c + o ( 1 ) ) 0 ( λ ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ13_HTML.gif
(3.8)
On the other hand, by the Hölder inequality and interpolation inequality, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ14_HTML.gif
(3.9)
By using the Gagliardo-Nirenberg inequality, we obtain
B R ε c ( 1 2 f ( g ( ( v n ) + ) ) g ( ( v n ) + ) F ( g ( ( v n ) + ) ) ) d x B R ε c ( 1 2 f ( g ( ( v n ) + ) ) g ( ( v n ) + ) C ( g ( ( v n ) + ) ) θ ) d x ( by  ( f 3 ) ) C B R c | g ( v n ) | p + 1 d x C ( B R c | g ( v n ) | 2 d x ) ( p + 1 ) β 2 ( B R c g 2 ( v n ) d x ) ( p + 1 ) ( 1 β ) 2 ( β = N ( p 1 ) 2 ( p + 1 ) ) C v n λ ( p + 1 ) β ( A ( R ) g 2 ( v n ) d x + B ( R ) g 2 ( v n ) d x ) ( p + 1 ) ( 1 β ) 2 C ( A ( R ) g 2 ( v n ) d x + B ( R ) g 2 ( v n ) d x ) ( p + 1 ) ( 1 β ) 2 ( v n λ  is bounded ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equy_HTML.gif

Let λ and R be large enough, from (3.8) and (3.9), we get the desired result. □

Lemma 3.8 c λ = inf N λ Φ λ ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq111_HTML.gif is achieved by some v > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq83_HTML.gif.

Proof By the definition of c λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq58_HTML.gif and the Ekeland variational principle, there exists a (PS) c sequence { v n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq75_HTML.gif, by Lemma 3.4, we know that { v n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq75_HTML.gif is bounded. Hence (up to a subsequence) we have v n v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq112_HTML.gif in L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq113_HTML.gif, v n v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq114_HTML.gif in L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq115_HTML.gif, v n v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq69_HTML.gif a.e. in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq15_HTML.gif , g ( v n ) g ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq116_HTML.gif in L q http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq117_HTML.gif for 2 q 2 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq118_HTML.gif.

It is sufficient to prove that v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq119_HTML.gif and v N λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq120_HTML.gif. In fact,
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ15_HTML.gif
(3.10)
it follows that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equz_HTML.gif
Let ε 0 = 1 4 c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq121_HTML.gif, since g ( v n ) g ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq122_HTML.gif strongly in L p + 1 ( B R ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq123_HTML.gif for R > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq110_HTML.gif, by Proposition 3.7, there exist Λ 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq124_HTML.gif, R 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq125_HTML.gif such that for λ Λ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq126_HTML.gif, R R 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq127_HTML.gif,
B R c | ( 1 2 f ( g ( ( v n ) + ) ) g ( ( v n ) + ) F ( g ( ( v n ) + ) ) ) ( 1 2 f ( g ( v + ) ) g ( v + ) F ( g ( v + ) ) ) | d x < ε 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equaa_HTML.gif
thus
B R ( 1 2 f ( g ( v + ) ) g ( v + ) F ( g ( v + ) ) ) d x 1 2 c > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equab_HTML.gif

Hence v 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq119_HTML.gif.

Now we prove v N λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq120_HTML.gif. Indeed, since { v n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq75_HTML.gif is a (PS) c sequence, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ16_HTML.gif
(3.11)

where o ( v n λ ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq128_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq129_HTML.gif.

Let l n : = g ( v n ) 1 + g 2 ( v n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq130_HTML.gif, then { l n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq131_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq67_HTML.gif for 2 q 2 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq132_HTML.gif, by the continuity of g, we have, up to a subsequence, l n l = g ( v ) 1 + g 2 ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq133_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq67_HTML.gif .

Similarly, we have s n : = f ( g ( v n ) ) 1 + g 2 ( v n ) f ( g ( v n ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq134_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq135_HTML.gif . Again, by the continuity of g, we have s n s = f ( g ( v ) ) 1 + g 2 ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq136_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq135_HTML.gif . Passing to the limits in (3.11), we get
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equac_HTML.gif

which is equivalent to Φ λ ( v ) , v = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq137_HTML.gif, that is, v N λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq120_HTML.gif. □

4 Proof of the main result

Consider the following quasilinear Schrödinger equation in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq138_HTML.gif ( N 3 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq2_HTML.gif):
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equad_HTML.gif
We have the same change of variables and the same notation as in the previous sections. Define the corresponding Orlicz space E G ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq139_HTML.gif by
E G ( Ω ) = { v | Ω g 2 ( v ) d x < + } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equae_HTML.gif
with the norm
| v | G ( Ω ) : = inf ξ > 0 ξ ( 1 + Ω G ( ξ 1 v ) d x ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equaf_HTML.gif
The space H G ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq140_HTML.gif is defined by
H G ( Ω ) : = { v | Ω | v | 2 d x < + , Ω g 2 ( v ) d x < + } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equag_HTML.gif
with the norm
v Ω = v L 2 + | v | G ( Ω ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equah_HTML.gif

The following Lemma 4.1 is a counterpart of Lemma 3.1.

Lemma 4.1 There exist two constants C 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq63_HTML.gif, C 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq64_HTML.gif such that
C 1 min { v Ω , v Ω 2 } Ω | v | 2 d x + Ω g 2 ( v ) d x C 2 max { v Ω , v Ω 2 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equai_HTML.gif

for any v H G ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq141_HTML.gif.

We denote by H G 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq142_HTML.gif the closure of C 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq143_HTML.gif in H G ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq140_HTML.gif. We define the functional Φ Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq144_HTML.gif on H G 0 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq142_HTML.gif by
Φ Ω ( v ) = 1 2 Ω ( | v | 2 + g 2 ( v ) ) d x Ω F ( g ( v + ) ) d x , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ17_HTML.gif
(4.1)
and we define the Nehari manifold N Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq145_HTML.gif by
N Ω : = { v H G ( Ω ) { 0 } | Φ Ω ( v ) , v = 0 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equaj_HTML.gif
Let
c ( Ω ) = inf N Ω Φ Ω ( v ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equak_HTML.gif

We recall that u = g ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq45_HTML.gif is a least energy solution of (D) if v N Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq146_HTML.gif such that c ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq147_HTML.gif is achieved.

Lemma 4.2 Suppose c λ = inf N λ Φ λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq148_HTML.gif. Then lim λ + c λ = c ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq149_HTML.gif.

Proof It is easy to see that c λ c ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq150_HTML.gif for λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq88_HTML.gif. We claim that c λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq58_HTML.gif is monotone increasing with respect to λ. In fact, for λ 1 < λ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq151_HTML.gif, we assume that c λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq152_HTML.gif, c λ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq153_HTML.gif are achieved for v λ 1 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq154_HTML.gif, v λ 2 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq155_HTML.gif. Obviously,
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ18_HTML.gif
(4.2)
We first prove that there exists 0 < α < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq156_HTML.gif such that α v λ 2 N λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq157_HTML.gif. This is sufficient to prove that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equal_HTML.gif
That is,
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equam_HTML.gif
Let
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equan_HTML.gif
Then by ( f 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq19_HTML.gif), we can obtain lim α 0 I ( α ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq158_HTML.gif and
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equao_HTML.gif
Hence, there exists 0 < α 0 < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq159_HTML.gif such that http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq160_HTML.gif , i.e., α 0 v λ 2 N λ 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq161_HTML.gif. Thus
c λ 1 Φ λ 1 ( α 0 v λ 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equap_HTML.gif
In the following, we will prove that
Φ λ 1 ( α 0 v λ 2 ) Φ λ 2 ( v λ 2 ) = c λ 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equaq_HTML.gif
In fact, we consider the function ρ ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq162_HTML.gif defined by
ρ ( α ) = ( f ( g ( α v λ 2 ) ) ( λ 2 a + 1 ) g ( α v λ 2 ) ) v λ 2 α 1 + g 2 ( α v λ 2 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equar_HTML.gif
By g ( t ) 1 + g 2 ( t ) 2 t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq163_HTML.gif for t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq164_HTML.gif, we have g ( α v λ 2 ) g ( α v λ 2 ) = g ( α v λ 2 ) 1 + g 2 ( α v λ 2 ) 2 α v λ 2 ( θ 2 ) α v λ 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq165_HTML.gif. It follows that
α v λ 2 g ( α v λ 2 ) g ( α v λ 2 ) / g ( α v λ 2 ) + α v λ 2 α v λ 2 g ( α v λ 2 ) ( θ 2 ) α v λ 2 + α v λ 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equas_HTML.gif
Obviously,
f ( g ( α v λ 2 ) ) g ( α v λ 2 ) v λ 2 2 α g ( α v λ 2 ) f ( g ( α v λ 2 ) ) v λ 2 ( g ( α v λ 2 ) + α g ( α v λ 2 ) v λ 2 ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equat_HTML.gif
and hence it is easy to check that
d d α ( f ( g ( α v λ 2 ) ) v λ 2 α 1 + g 2 ( α v λ 2 ) ) = d d α ( g ( α v λ 2 ) 1 + g 2 ( α v λ 2 ) ) f ( g ( α v λ 2 ) ) v λ 2 α g ( α v λ 2 ) + d d α ( f ( g ( α v λ 2 ) ) v λ 2 α g ( α v λ 2 ) ) g ( α v λ 2 ) 1 + g 2 ( α v λ 2 ) = v λ 2 2 ( 1 + g 2 ( α v λ 2 ) ) 2 f ( g ( α v λ 2 ) ) v λ 2 α g ( α v λ 2 ) + f ( g ( α v λ 2 ) ) g ( α v λ 2 ) v λ 2 2 α g ( α v λ 2 ) f ( g ( α v λ 2 ) ) v λ 2 ( g ( α v λ 2 ) + α g ( α v λ 2 ) v λ 2 ) α 2 g 2 ( α v λ 2 ) × g ( α v λ 2 ) 1 + g 2 ( α v λ 2 ) f ( g ( α v λ 2 ) ) g ( α v λ 2 ) v λ 2 2 α g ( α v λ 2 ) f ( g ( α v λ 2 ) ) v λ 2 ( g ( α v λ 2 ) + α g ( α v λ 2 ) v λ 2 ) α 2 g ( α v λ 2 ) 1 + g 2 ( α v λ 2 ) 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equau_HTML.gif
On the other hand,
d d α ( ( λ 2 a + 1 ) g ( α v λ 2 ) v λ 2 α 1 + g 2 ( α v λ 2 ) ) = ( λ 2 a + 1 ) α v λ 2 2 g ( α v λ 2 ) 1 + g 2 ( α v λ 2 ) ( 1 + g 2 ( α v λ 2 ) ) α 2 ( 1 + g 2 ( α v λ 2 ) ) 2 v λ 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equav_HTML.gif
by v = 1 2 g ( v ) 1 + g 2 ( v ) + 1 2 ln ( g ( v ) + 1 + g 2 ( v ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq166_HTML.gif, it is easy to check that for any t 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq164_HTML.gif,
t g ( t ) 1 + g 2 ( t ) ( 1 + g 2 ( t ) ) 0 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equaw_HTML.gif
which implies
d d α ( ( λ 2 a + 1 ) g ( α v λ 2 ) v λ 2 α 1 + g 2 ( α v λ 2 ) ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equax_HTML.gif

for any α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq167_HTML.gif, thus we have proved that ρ ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq162_HTML.gif is monotone increasing for α > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq167_HTML.gif.

Now we consider the function γ ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq168_HTML.gif defined by
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equay_HTML.gif
Then
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equaz_HTML.gif
for 0 < α < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq156_HTML.gif. Therefore, γ ( α ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq168_HTML.gif is monotone increasing with respect to α < 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq169_HTML.gif. Thus, we deduce that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equba_HTML.gif

Assume that lim λ c λ = k c ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq170_HTML.gif. If k < c ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq171_HTML.gif, then for any sequence { λ n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq172_HTML.gif ( λ n + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq173_HTML.gif), we have c λ n k < c ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq174_HTML.gif.

We assume that v n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq175_HTML.gif is such that c λ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq176_HTML.gif is achieved, by Lemma 3.4, { v n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq75_HTML.gif is bounded in H G λ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq177_HTML.gif. Since v n H G 0 v n H G λ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq178_HTML.gif, { v n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq75_HTML.gif is bounded in H G 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq179_HTML.gif, as a result, we have v n v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq180_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq181_HTML.gif , g ( v n ) g ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq122_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq182_HTML.gif for 2 q 2 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq132_HTML.gif, v n v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq112_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq67_HTML.gif for 2 q 2 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq132_HTML.gif, v n v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq69_HTML.gif a.e. in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq15_HTML.gif .

We claim that v | Ω c = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq183_HTML.gif, where http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq184_HTML.gif . Indeed, it is sufficient to prove g ( v ) | Ω c = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq185_HTML.gif. If not, then there exists a compact subset Σ Ω c http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq186_HTML.gif with dist { Σ , Ω } > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq187_HTML.gif such that g ( v ) | Σ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq188_HTML.gif and
Σ g 2 ( v n ) d x Σ g 2 ( v ) d x > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbb_HTML.gif

Moreover, there exists ε 0 > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq189_HTML.gif such that a ( x ) ε 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq190_HTML.gif for any x Σ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq191_HTML.gif.

By the choice of v n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq175_HTML.gif, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbc_HTML.gif
hence,
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbd_HTML.gif

This contradiction shows that g ( v ) | Ω c = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq185_HTML.gif and so does v.

Now we show that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ19_HTML.gif
(4.3)
Suppose that (4.3) is not true, then by the concentration compactness principle of Lions (see [12]), there exist δ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq192_HTML.gif, ϱ > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq193_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq194_HTML.gif with | x n | + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq195_HTML.gif such that
lim sup n B ϱ ( x n ) | g ( v n ) g ( v ) | 2 δ > 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Eqube_HTML.gif
On the other hand, by the choice of { v n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq75_HTML.gif, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbf_HTML.gif

which shows that g ( v n ) g ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq122_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq67_HTML.gif for 2 < q < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq196_HTML.gif. In the above proof, we have used the fact that μ { B ϱ ( x n ) { x | a ( x ) M 0 } } 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq197_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq129_HTML.gif and the L 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq115_HTML.gif bounded property of g ( v n ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq198_HTML.gif.

Now, since { v n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq75_HTML.gif is bounded, by the Fatou lemma, we obtain
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbg_HTML.gif
But, by the choice of v n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq175_HTML.gif, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbh_HTML.gif
hence,
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equ20_HTML.gif
(4.4)
In the following, we will prove that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbi_HTML.gif
Indeed,
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbj_HTML.gif
Since f ( g ( v + ) ) g ( ( v n ) + ) v n f ( g ( v + ) ) g ( v + ) v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq199_HTML.gif, one can easily see that I 2 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq200_HTML.gif as n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq129_HTML.gif, and
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbk_HTML.gif
by using g ( v n ) g ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq122_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq67_HTML.gif for 2 < q < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq196_HTML.gif. It follows from (4.4) that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbl_HTML.gif
thus, there exists 0 < α 0 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq201_HTML.gif such that α 0 v N Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq202_HTML.gif and
Φ Ω ( α 0 v ) Φ Ω ( v ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbm_HTML.gif

hence c ( Ω ) Φ Ω ( α 0 v ) < Φ Ω ( v ) lim n Φ λ n = k < c ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq203_HTML.gif. A contradiction. Thus we have proved that lim λ c λ c ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq204_HTML.gif as λ + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq205_HTML.gif. □

Proof of Theorem 2.1 Suppose that { v n } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq75_HTML.gif is a sequence such that v n N λ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq206_HTML.gif, Φ λ n ( v n ) = c λ n http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq207_HTML.gif, by the proof of Lemma 3.2, we have v n v http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq180_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq181_HTML.gif , g ( v n ) g ( v ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq122_HTML.gif in http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq67_HTML.gif for 2 < q < 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq196_HTML.gif and v | Ω c = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq183_HTML.gif. Moreover, Φ Ω ( v ) lim n Φ λ n ( v n ) c ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq208_HTML.gif, and if v N Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq146_HTML.gif, then Φ Ω ( v ) = c ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq209_HTML.gif. Hence, in the following, we need only to prove that v N Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq146_HTML.gif. To do this, it is sufficient to prove that
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbn_HTML.gif
and
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbo_HTML.gif
In fact, if one of the above three limits does not hold, by the Fatou lemma, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_Equbp_HTML.gif

Similar to above, there exists α 0 ( 0 , 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq210_HTML.gif such that α 0 v N Ω http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq202_HTML.gif and c ( Ω ) Φ Ω ( α 0 v ) < Φ Ω ( v ) lim n Φ λ n ( v n ) c ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-9/MediaObjects/13661_2012_Article_264_IEq211_HTML.gif. A contradiction, and thus we complete the proof of Theorem 2.1. □

Declarations

Acknowledgements

The author would like to thank the referee for some valuable comments and helpful suggestions. This study was supported by the National Natural Science Foundation of China (11161041, 31260098) and the Fundamental Research Funds for the Central Universities (zyz2012080, zyz2012074).

Authors’ Affiliations

(1)
College of Mathematics and Computer Science, Northwest University for Nationalities
(2)
College of Mathematics and Statistics, Northwest Normal University

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