SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Least energy solutions for a quasilinear Schrödinger equation with potential well

Yujuan Jiao

Author affiliations

College of Mathematics and Computer Science, Northwest University for Nationalities, Lanzhou, 730124, P.R. China

College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, P.R. China

Citation and License

Boundary Value Problems 2013, 2013:9  doi:10.1186/1687-2770-2013-9

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/9


Received:23 October 2012
Accepted:5 January 2013
Published:21 January 2013

© 2013 Jiao; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M2">View MathML</a> having a potential well, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M3">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M4">View MathML</a> is a parameter. Under suitable hypotheses, we obtain the existence of a least energy solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M5">View MathML</a> of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M6">View MathML</a>) which localizes near the potential well <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M7">View MathML</a> 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:

for sufficiently large λ, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M3">View MathML</a>.

Our assumptions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M10">View MathML</a> are as follows:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M11">View MathML</a>) , the potential well <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M13">View MathML</a> is a non-empty set and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M14">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M15">View MathML</a>) There exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M16">View MathML</a> such that , where μ denotes the Lebesgue measure on .

Condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M15">View MathML</a>) is very weak in dealing with the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M20">View MathML</a> on , which was firstly used by Bartsch and Wang [1] in dealing with the semilinear Schrödinger equation.

Remark 1.1<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M22">View MathML</a> can be unbounded.

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M23">View MathML</a>, we assume that f is continuous and satisfies the following conditions:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M24">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M25">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M26">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M27">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M28">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M29">View MathML</a> is a constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M30">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M31">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M32">View MathML</a>) There is a number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M33">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M34">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M35">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M36">View MathML</a>.

Hypotheses (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M11">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M15">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M24">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M26">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M32">View MathML</a>) will be maintained throughout this paper.

Solutions of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M6">View MathML</a>) are related to the existence of the standing wave solutions of the following quasilinear Schrödinger equation:

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M44">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M45">View MathML</a> was used for the superfluid film equation in plasma physics by Kurihara [2] (see also [3]); in the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M46">View MathML</a>, (1.1) was used as a model of the self-changing of a high-power ultrashort laser in matter (see [4-7] and references therein).

In recent years, much attention has been devoted to the quasilinear Schrödinger equation of the following form:

(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 , 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:

(1.3)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M2">View MathML</a> having a potential well and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M30">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M31">View MathML</a> is the critical Sobolev exponent, and they proved the existence of a ground state solution of (1.3) which localizes near the potential well <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M53">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M54">View MathML</a> 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 (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M6">View MathML</a>), the existence of least energy solutions for equation (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M6">View MathML</a>) with a potential well <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M53">View MathML</a> for λ large is proved under the conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M11">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M15">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M24">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M26">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M32">View MathML</a>).

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M63">View MathML</a>. Formally, we define the following functional:

(2.1)

for . Note that under our assumptions, the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M66">View MathML</a> is not well defined on X.

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

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M67">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M68">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M69">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M70">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M69">View MathML</a> is strictly monotone and hence has an inverse function denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M73">View MathML</a>. Obviously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M74">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M75">View MathML</a>. Then it holds that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M76">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M77">View MathML</a> is convex. Moreover, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M78">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M79">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M80">View MathML</a>

Now we introduce the Orlicz space (see [26])

equipped with the norm

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M83">View MathML</a> is a Banach space.

Let

equipped with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M85">View MathML</a>

Using the change of variable, we define the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M87">View MathML</a> by

(2.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M89">View MathML</a> is the positive part of v.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M90">View MathML</a>

be the Nehari manifold and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M91">View MathML</a>

be the infimum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86">View MathML</a> on the Nehari manifold <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M93">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M94">View MathML</a> is the Gateaux derivative (see Proposition 3.3).

We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M95">View MathML</a> is a least energy solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M6">View MathML</a>) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M97">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M98">View MathML</a> is achieved.

Note that under our assumptions, for λ large enough, the following Dirichlet problem is a kind of a ‘limit’ problem:

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M100">View MathML</a>.

Similar to the definition of the least energy solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M6">View MathML</a>), we can define the least energy solution of (D) which will be given in Section 4.

Our main result is as follows.

Theorem 2.1Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M11">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M15">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M24">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M26">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M32">View MathML</a>) are satisfied. Then forλlarge, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M98">View MathML</a>is achieved by a critical point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M108">View MathML</a>of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M95">View MathML</a>is a least energy solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M6">View MathML</a>). Furthermore, for any sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M113">View MathML</a>has a subsequence converging tovsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M73">View MathML</a>is a least energy solution of (D).

3 Preliminaries

In order to obtain the compactness of the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86">View MathML</a>, we recall the following Lemmas 3.1 and 3.2 which can be found in [13].

Lemma 3.1There exist two constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M117">View MathML</a>such that

(3.1)

for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M119">View MathML</a>.

Lemma 3.2The map: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M120">View MathML</a>from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M87">View MathML</a>intois continuous for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M123">View MathML</a>.

Now we consider the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86">View MathML</a> defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M87">View MathML</a> by (2.2), the following Proposition 3.3 is due to [9].

Proposition 3.3

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86">View MathML</a>is well defined on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M87">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86">View MathML</a>is continuous in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M87">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86">View MathML</a>is Gateaux differentiable, the Gateaux derivative<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M94">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M119">View MathML</a>is a linear functional and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M94">View MathML</a>is continuous invin the strong-weak topology, that is, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M134">View MathML</a>strongly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M87">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M136">View MathML</a>weakly. Moreover, the Gateaux derivative<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M94">View MathML</a>has the form

(3.2)

Recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M139">View MathML</a> is called a Palais-Smale sequence ((PS)c sequence in short) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M141">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M142">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M143">View MathML</a>, the dual space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M87">View MathML</a>. We say that the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86">View MathML</a> satisfies the (PS)c condition if any of (PS)c sequence (up to a subsequence, if necessary) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146">View MathML</a> converges strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M87">View MathML</a>.

Lemma 3.4Any of (PS)csequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86">View MathML</a>is bounded.

Proof Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146">View MathML</a> is a (PS)c sequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86">View MathML</a>. We have and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M142">View MathML</a> in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M143">View MathML</a>.

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M155">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M156">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M157">View MathML</a>, thus

(3.3)

and

(3.4)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M160">View MathML</a> yields

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M162">View MathML</a>

we have

It follows from Lemma 3.1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M164">View MathML</a>

(3.5)

thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M87">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M167">View MathML</a> be the critical set of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86">View MathML</a>. Suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M169">View MathML</a>, then it is easy to check that either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M170">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M171">View MathML</a> in by the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86">View MathML</a> and the strong maximum principle. □

Lemma 3.5There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M174">View MathML</a>which is independent ofλsuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M175">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M176">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M177">View MathML</a>.

Proof Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M178">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M176">View MathML</a> (otherwise, the conclusion is true). From (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M24">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M26">View MathML</a>), we see that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M182">View MathML</a>, there is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M183">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M184">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M34">View MathML</a>. We have

and we can easily deduce the desired result. □

Lemma 3.6There exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M187">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M188">View MathML</a>

and either<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M189">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M190">View MathML</a>if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146">View MathML</a>is a (PS)csequence for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M193">View MathML</a>is the constant in Lemma 3.1.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146">View MathML</a> is a (PS)c sequence, we have

It follows from (3.5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M196">View MathML</a>

On the other hand, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M197">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M198">View MathML</a>

(3.6)

Thus, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M199">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M200">View MathML</a>) such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M201">View MathML</a>

(3.7)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M202">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M203">View MathML</a>

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M204">View MathML</a>. It follows from (3.6) and (3.7) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M205">View MathML</a>

hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M206">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M190">View MathML</a>. Therefore, we have proved that there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M208">View MathML</a> such that either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M189">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M190">View MathML</a>. □

Proposition 3.7Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M211">View MathML</a>be a constant. Then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M182">View MathML</a>, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M214">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M215">View MathML</a>

if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146">View MathML</a>is a (PS)csequence of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M86">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M219">View MathML</a>, where.

Proof For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M221">View MathML</a>, let

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M223">View MathML</a>

(3.8)

On the other hand, by the Hölder inequality and interpolation inequality, we have

(3.9)

By using the Gagliardo-Nirenberg inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M225">View MathML</a>

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

Lemma 3.8<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M226">View MathML</a>is achieved by some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M170">View MathML</a>.

Proof By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M98">View MathML</a> and the Ekeland variational principle, there exists a (PS)c sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146">View MathML</a>, by Lemma 3.4, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146">View MathML</a> is bounded. Hence (up to a subsequence) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M231">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M233">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M234">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M134">View MathML</a> a.e. in , <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M237">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M238">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M239">View MathML</a>.

It is sufficient to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M240">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M241">View MathML</a>. In fact,

(3.10)

it follows that

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M244">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M245">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M246">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M221">View MathML</a>, by Proposition 3.7, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M248">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M249">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M250">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M251">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M252">View MathML</a>

thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M253">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M240">View MathML</a>.

Now we prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M241">View MathML</a>. Indeed, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146">View MathML</a> is a (PS)c sequence, we have

(3.11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M258">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M259">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M260">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M261">View MathML</a> is bounded in for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M263">View MathML</a>, by the continuity of g, we have, up to a subsequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M264">View MathML</a> in .

Similarly, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M266">View MathML</a> is bounded in . Again, by the continuity of g, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M268">View MathML</a> in . Passing to the limits in (3.11), we get

which is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M271">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M241">View MathML</a>. □

4 Proof of the main result

Consider the following quasilinear Schrödinger equation in (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M3">View MathML</a>):

We have the same change of variables and the same notation as in the previous sections. Define the corresponding Orlicz space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M276">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M277">View MathML</a>

with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M278">View MathML</a>

The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M279">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M280">View MathML</a>

with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M281">View MathML</a>

The following Lemma 4.1 is a counterpart of Lemma 3.1.

Lemma 4.1There exist two constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M116">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M117">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M284">View MathML</a>

for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M285">View MathML</a>.

We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M286">View MathML</a> the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M287">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M279">View MathML</a>. We define the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M289">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M286">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M291">View MathML</a>

(4.1)

and we define the Nehari manifold <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M292">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M293">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M294">View MathML</a>

We recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M73">View MathML</a> is a least energy solution of (D) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M296">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M297">View MathML</a> is achieved.

Lemma 4.2Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M298">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M299">View MathML</a>.

Proof It is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M300">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M177">View MathML</a>. We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M98">View MathML</a> is monotone increasing with respect to λ. In fact, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M303">View MathML</a>, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M304">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M305">View MathML</a> are achieved for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M306">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M307">View MathML</a>. Obviously,

(4.2)

We first prove that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M309">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M310">View MathML</a>. This is sufficient to prove that

That is,

Let

Then by (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M24">View MathML</a>), we can obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M315">View MathML</a> and

Hence, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M317">View MathML</a> such that , i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M319">View MathML</a>. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M320">View MathML</a>

In the following, we will prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M321">View MathML</a>

In fact, we consider the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M322">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M323">View MathML</a>

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M324">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M325">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M326">View MathML</a>. It follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M327">View MathML</a>

Obviously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M328">View MathML</a>

and hence it is easy to check that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M329">View MathML</a>

On the other hand,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M330">View MathML</a>

by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M331">View MathML</a>, it is easy to check that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M325">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M333">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M334">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M335">View MathML</a>, thus we have proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M322">View MathML</a> is monotone increasing for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M335">View MathML</a>.

Now we consider the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M338">View MathML</a> defined by

Then

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M309">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M338">View MathML</a> is monotone increasing with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M343">View MathML</a>. Thus, we deduce that

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M345">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M346">View MathML</a>, then for any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M347">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M348">View MathML</a>), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M349">View MathML</a>.

We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M350">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M351">View MathML</a> is achieved, by Lemma 3.4, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M353">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M354">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M356">View MathML</a>, as a result, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M357">View MathML</a> in , <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M245">View MathML</a> in for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M263">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M231">View MathML</a> in for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M263">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M134">View MathML</a> a.e. in .

We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M367">View MathML</a>, where . Indeed, it is sufficient to prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M369">View MathML</a>. If not, then there exists a compact subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M370">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M371">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M372">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M373">View MathML</a>

Moreover, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M374">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M375">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M376">View MathML</a>.

By the choice of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M350">View MathML</a>, we have

hence,

This contradiction shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M369">View MathML</a> and so does v.

Now we show that

(4.3)

Suppose that (4.3) is not true, then by the concentration compactness principle of Lions (see [12]), there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M382">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M383">View MathML</a> and with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M385">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M386">View MathML</a>

On the other hand, by the choice of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146">View MathML</a>, we have

which shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M245">View MathML</a> in for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M391">View MathML</a>. In the above proof, we have used the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M392">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M259">View MathML</a> and the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M234">View MathML</a> bounded property of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M395">View MathML</a>.

Now, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146">View MathML</a> is bounded, by the Fatou lemma, we obtain

But, by the choice of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M350">View MathML</a>, we have

hence,

(4.4)

In the following, we will prove that

Indeed,

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M403">View MathML</a>, one can easily see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M404">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M259">View MathML</a>, and

by using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M245">View MathML</a> in for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M391">View MathML</a>. It follows from (4.4) that

thus, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M411">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M412">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M413">View MathML</a>

hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M414">View MathML</a>. A contradiction. Thus we have proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M415">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M416">View MathML</a>. □

Proof of Theorem 2.1 Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M146">View MathML</a> is a sequence such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M418">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M419">View MathML</a>, by the proof of Lemma 3.2, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M357">View MathML</a> in , <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M245">View MathML</a> in for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M391">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M367">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M426">View MathML</a>, and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M296">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M428">View MathML</a>. Hence, in the following, we need only to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M296">View MathML</a>. To do this, it is sufficient to prove that

and

In fact, if one of the above three limits does not hold, by the Fatou lemma, we have

Similar to above, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M433">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M412">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M435">View MathML</a>. A contradiction, and thus we complete the proof of Theorem 2.1. □

Competing interests

The author declares that she has no competing interests.

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).

References

  1. Bartsch, T, Wang, Z: Multiple positive solutions for a nonlinear Schrödinger equation. Z. Angew. Math. Phys.. 51, 366–384 (2000)

  2. Kurihura, S: Large-amplitude quasi-solitons in superfluid. J. Phys. Soc. Jpn.. 50, 3262–3267 (1981). Publisher Full Text OpenURL

  3. Laedke, EW, Spatschek, KH, Stenflo, L: Evolution theorem for a class of perturbed envelop soliton solutions. J. Math. Phys.. 24, 2764–2769 (1983). Publisher Full Text OpenURL

  4. Brandi, HS, Manus, C, Mainfray, G, Lehner, T, Bonnaud, G: Relativistic and ponderomotive self-focusing of a laser beam in a radially inhomogeneous plasma. Phys. Fluids B. 5, 3539–3550 (1993). Publisher Full Text OpenURL

  5. Chen, XL, Sudan, RN: Necessary and sufficient conditions for self-focusing of short ultraintense laser pulse. Phys. Rev. Lett.. 70, 2082–2085 (1993). PubMed Abstract | Publisher Full Text OpenURL

  6. De Bouard, A, Hayashi, N, Saut, JC: Global existence of small solutions to a relativistic nonlinear Schrödinger equation. Commun. Math. Phys.. 189, 73–105 (1997). Publisher Full Text OpenURL

  7. Ritchie, B: Relativistic self-focusing channel formation in laser-plasma interactions. Phys. Rev. E. 50, 687–689 (1994). Publisher Full Text OpenURL

  8. Poppenberg, M, Schmitt, K, Wang, Z: On the existence of solutions to quasilinear Schrödinger equation. Calc. Var. Partial Differ. Equ.. 14, 329–344 (2002). Publisher Full Text OpenURL

  9. Liu, J, Wang, Y, Wang, Z: Soliton solutions for quasilinear Schrödinger equation II. J. Differ. Equ.. 187, 473–493 (2003). Publisher Full Text OpenURL

  10. Colin, M, Jeanjean, L: Solutions for a quasilinear Schrödinger equation: a dual approach. Nonlinear Anal.. 56, 213–226 (2004). Publisher Full Text OpenURL

  11. Berestycki, H, Lions, PL: Nonlinear scalar field equations I. Arch. Ration. Mech. Anal.. 82, 313–346 (1983)

  12. Lions, PL: The concentration-compactness principle in the calculus of variations. The locally compact case. Part I. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 1, 109–145 (1984)

  13. Guo, Y, Tang, Z: Ground state solutions for the quasilinear Schrödinger equation. Nonlinear Anal.. 75, 3235–3248 (2012). Publisher Full Text OpenURL

  14. Guo, Y, Tang, Z: Ground state solutions for the quasilinear Schrödinger systems. J. Math. Anal. Appl.. 389, 322–339 (2012). Publisher Full Text OpenURL

  15. Colin, M, Jeanjean, L, Squassina, M: Stability and instability results for standing waves of quasi-linear Schrödinger equations. Nonlinearity. 23, 1353–1385 (2010). Publisher Full Text OpenURL

  16. Ambrosetti, A, Badiale, M, Cingolani, S: Semiclassical states of nonlinear Schrödinger equations. Arch. Ration. Mech. Anal.. 140, 285–300 (1997). Publisher Full Text OpenURL

  17. Ambrosetti, A, Malchiodi, A, Secchi, S: Multiplicity results for some nonlinear Schrödinger equations with potentials. Arch. Ration. Mech. Anal.. 159, 253–271 (2001). Publisher Full Text OpenURL

  18. Byeon, J, Wang, Z: Standing waves with a critical frequency for nonlinear Schrödinger equations II. Calc. Var. Partial Differ. Equ.. 18, 207–219 (2003). Publisher Full Text OpenURL

  19. Cingolani, S, Lazzo, M: Multiple positive solutions to nonlinear Schrödinger equations with competing potential functions. J. Differ. Equ.. 160, 118–138 (2000). Publisher Full Text OpenURL

  20. Cingolani, S, Nolasco, M: Multi-peaks periodic semiclassical states for a class of nonlinear Schrödinger equations. Proc. R. Soc. Edinb.. 128, 1249–1260 (1998). Publisher Full Text OpenURL

  21. Del Pino, M, Felmer, P: Semi-classical states for nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré. 15, 127–149 (1998)

  22. Del Pino, M, Felmer, P: Multi-peak bound states for nonlinear Schrödinger equations. J. Funct. Anal.. 149, 245–265 (1997). Publisher Full Text OpenURL

  23. Floer, A, Weinstein, A: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. J. Funct. Anal.. 69, 397–408 (1986). PubMed Abstract | Publisher Full Text OpenURL

  24. Oh, YG: On positive multi-bump bound states of nonlinear Schrödinger equations under multiple well potential. Commun. Math. Phys.. 131, 223–253 (1990). Publisher Full Text OpenURL

  25. Oh, YG: Existence of semiclassical bound states of nonlinear Schrödinger equations with potentials of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/9/mathml/M436">View MathML</a>. Commun. Partial Differ. Equ.. 13, 1499–1519 (1988). Publisher Full Text OpenURL

  26. Rao, M, Ren, Z: Theory of Orlicz Space, Dekker, New York (1991)