SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

Open Access Research

Multiple solutions to nonlinear Schrödinger equations with critical growth

Wulong Liu1* and Peihao Zhao2

Author Affiliations

1 Department of Mathematics, Jiangxi University of Science and Technology, Ganzhou, 341000, P.R. China

2 School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, P.R. China

For all author emails, please log on.

Boundary Value Problems 2013, 2013:199  doi:10.1186/1687-2770-2013-199

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/199


Received:20 March 2013
Accepted:24 May 2013
Published:4 September 2013

© 2013 Liu and Zhao; 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 2000, Cingolani and Lazzo (J. Differ. Equ. 160:118-138, 2000) studied nonlinear Schrödinger equations with competing potential functions and considered only the subcritical growth. They related the number of solutions with the topology of the global minima set of a suitable ground energy function. In the present paper, we establish these results in the critical case. In particular, we remove the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M1">View MathML</a>, which is a key condition in their paper. In the proofs we apply variational methods and Ljusternik-Schnirelmann theory.

MSC: 35J60, 35Q55.

Keywords:
nonlinear Schrödinger equations; critical growth; variational methods

1 Introduction and main result

We investigate the following nonlinear Schrödinger equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M2">View MathML</a>

(1.1)

which arises in quantum mechanics and provides a description of the dynamics of the particle in a non-relativistic setting. ħ is the Planck’s constant, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M3">View MathML</a> denotes the mass of the particle, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M4">View MathML</a> is the electric potential, g is the nonlinear coupling, and ψ is the wave function representing the state of the particle. A standing wave solution of equation (1.1) is a solution of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M5">View MathML</a>. It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M6">View MathML</a> solves (1.1) if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M7">View MathML</a> solves the following stationary equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M8">View MathML</a>

(1.2)

For simplicity and without loss of generality, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M11">View MathML</a>, then equation (1.2) is equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M12">View MathML</a>

(1.3)

A considerable amount of work has been devoted to investigating solutions of (1.3). The existence, multiplicity and qualitative property of such solutions have been extensively studied. For single interior spikes solutions in the whole space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M13">View MathML</a>, please see [1-9]etc. For multiple interior spikes, please see [10,11]etc. For single boundary spike solutions with Neumann boundary condition, please see [6,12-15]etc. For multiple boundary spikes, please see [16-18]etc. In particular, Wang and Zeng [9] studied the existence and concentration behavior of solutions for NLS with competing potential functions. Cingolani and Lazzo in [19] obtained the multiple solutions for the similar equation. In those papers only the subcritical growth was considered. In the present paper, we complete these studies by considering a class of nonlinearities with the critical growth. In particular, we remove the condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M1">View MathML</a>, which is a key condition in [19].

In the sequel, we restrict ourselves to the critical case in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M15">View MathML</a>. More specifically, we study the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M16">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M17">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M18">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M19">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M20">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M21">View MathML</a> satisfies

(f1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M22">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M23">View MathML</a>;

(f2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M24">View MathML</a>;

(f3) there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M25">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M26">View MathML</a>

(f4) there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M27">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M28">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M29">View MathML</a>;

(f5) the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M30">View MathML</a> is strictly increasing in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M31">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M32">View MathML</a>.

Our main results are the following theorem.

Theorem 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M33">View MathML</a>. Suppose that f satisfies (f1)-(f5), Vis a continuous function in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M13">View MathML</a>and satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M35">View MathML</a>. Then whenεis sufficiently small, the problem (1.4) has at least<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M36">View MathML</a>distinct nontrivial solutions.

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M36">View MathML</a> denotes the Ljusternik-Schnirelmann category of Σ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M38">View MathML</a>. By definition (e.g., [20]), the category of A with respect to M, denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M39">View MathML</a>, is the least integer k such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M40">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M41">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M42">View MathML</a>) closed and contractible in M. We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M43">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M44">View MathML</a> if there are no integers with the above property. We will use the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M45">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M46">View MathML</a>.

To prove Theorem 1.1, we mainly use the idea of [15,19,21]. More precisely, we can show that the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M47">View MathML</a>-condition holds in the subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M48">View MathML</a> (see (4.6)). Hence the standard Ljusternik-Schnirelmann category theory can be applied in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M48">View MathML</a> to yield the existence of at least <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M50">View MathML</a> critical points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51">View MathML</a>. And then we construct two continuous mappings

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M52">View MathML</a>

(1.5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M53">View MathML</a>

(1.6)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M54">View MathML</a>

(1.7)

Then a topological argument asserts that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M55">View MathML</a>

We will also prove that if u is a critical point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M57">View MathML</a>, then u cannot change sign. Hence we obtain at least <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M58">View MathML</a> nontrivial critical points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51">View MathML</a>.

The paper is organized as follows. In Section 2, we collect some notations and preliminaries. A compactness result is given in Section 3, which is a key step in our proof. Finally, in Section 4, we prove Theorem 1.1.

2 Notations and preliminaries

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60">View MathML</a> is the usual Sobolev space of real-valued functions defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M61">View MathML</a>

with the normal

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M62">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M63">View MathML</a> be the subspace of a Hilbert space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60">View MathML</a> with respect to the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M65">View MathML</a>

We denote by S the Sobolev constant for the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M66">View MathML</a>, namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M67">View MathML</a>

(2.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M68">View MathML</a> is the usual Sobolev space of real-valued functions defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M69">View MathML</a>

We say that a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M70">View MathML</a> is a weak solution of the problem (1.4) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M71">View MathML</a>

In view of (f2) and (f3), we have that the associated functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M72">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M73">View MathML</a>

is well defined. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M74">View MathML</a> with the following derivative:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M75">View MathML</a>

Hence, the weak solutions of (1.4) are exactly the critical points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51">View MathML</a>.

Let us recall some known facts about the limiting problem, namely the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M77">View MathML</a>

(2.2)

here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M78">View MathML</a> acts as a parameter instead of an independent variable. Solutions of (2.2) will be sought in the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60">View MathML</a> as critical points of the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M80">View MathML</a>

The least positive critical value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M81">View MathML</a> can be characterized as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M82">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M83">View MathML</a>

(2.3)

An associated critical point w actually solves equation (2.2) and is called a ground state solution or the least energy solution, i.e., w satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M84">View MathML</a>

Moreover, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M85">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M86">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M87">View MathML</a>

(2.4)

For more details, please see [22,23].

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M88">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M86">View MathML</a>, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M90">View MathML</a>. We need to estimate the super bound of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M91">View MathML</a>. In order to do this, we estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M81">View MathML</a>. We shall use a family of radial function defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M93">View MathML</a>

It is known [20] that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M94">View MathML</a>

Moreover, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M95">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M96">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M97">View MathML</a> is a cut-off function satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M98">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M100">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M101">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M102">View MathML</a>. After a detailed calculation, we have the following estimates:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M103">View MathML</a>

(2.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M104">View MathML</a>

(2.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M105">View MathML</a>

(2.7)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M106">View MathML</a>, from (2.5)-(2.7), we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M107">View MathML</a>

(2.8)

then the maximum value of the right-hand side is achieved at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M108">View MathML</a>

(2.9)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M109">View MathML</a>

(2.10)

Hence we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M110">View MathML</a>

(2.11)

We denote the Nehari manifold of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M112">View MathML</a>

3 Compactness result

Proposition 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M113">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M115">View MathML</a>satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M116">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M117">View MathML</a>. Then uniformly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M118">View MathML</a>, there exist a subsequence of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M119">View MathML</a> (still denoted by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M120">View MathML</a>), and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M121">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M122">View MathML</a>. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M123">View MathML</a>converges strongly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60">View MathML</a>tow, the positive ground state solution of equation (2.2).

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M125">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M126">View MathML</a>. Then, by a change of variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M127">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M128">View MathML</a>

(3.1)

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M129">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60">View MathML</a>. Noting that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M131">View MathML</a>

(3.2)

hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M132">View MathML</a>

(3.3)

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M133">View MathML</a>. Now we prove that there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M134">View MathML</a> and constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M135">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M136">View MathML</a>

(3.4)

Indeed, if this is not true, then the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M129">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60">View MathML</a> and a lemma due to Lions [[24], Lemma I.1] imply that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M139">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M140">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M141">View MathML</a>. Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M86">View MathML</a>, we can use (f2), (f3) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M143">View MathML</a> to get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M144">View MathML</a>

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M145">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114">View MathML</a>. Therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M147">View MathML</a>

(3.5)

and consequently (3.2) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M148">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M149">View MathML</a>

(3.6)

However, recall the definition of S in (2.1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M150">View MathML</a>

equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M151">View MathML</a>, contradicting (3.6). Thus, (3.4) holds. Using the idea of [21,25], along a subsequence as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114">View MathML</a>, we may assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M153">View MathML</a>

We now consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M121">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M122">View MathML</a> (see (2.3)). By a change of variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M127">View MathML</a>, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M157">View MathML</a>

(3.7)

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M158">View MathML</a>, from which it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M159">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M161">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M162">View MathML</a> are bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M164">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60">View MathML</a>, the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M166">View MathML</a> is bounded. Thus, up to a subsequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M167">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M168">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M169">View MathML</a>, which does not occur. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M170">View MathML</a>, and therefore the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M162">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M172">View MathML</a>

(3.8)

For fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M173">View MathML</a>, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M174">View MathML</a>

By the Hölder inequality,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M175">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M176">View MathML</a>, the dual space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60">View MathML</a>. Consequently, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M179">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M181">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M182">View MathML</a>

(3.9)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M123">View MathML</a> converges weakly to w in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M123">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M186">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M187">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M188">View MathML</a>. It then follows that there is a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M162">View MathML</a>, still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M162">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M187">View MathML</a> converges weakly to some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M192">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M188">View MathML</a>. Next we will show <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M194">View MathML</a>. Choose a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M195">View MathML</a> of open relatively compact subsets, with regular boundaries, of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M13">View MathML</a> covering <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M13">View MathML</a>, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M198">View MathML</a>. It is easy to see that, by compact embedding, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M199">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M200">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M201">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M199">View MathML</a> a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M203">View MathML</a>. Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M204">View MathML</a> a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M203">View MathML</a>. By the Brezis and Lieb lemma [26], we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M206">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M207">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M194">View MathML</a> a.e. on each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M203">View MathML</a>, and then the diagonal rule implies a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M13">View MathML</a>. Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M211">View MathML</a>

(3.10)

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M212">View MathML</a>

(3.11)

By (f2) and (f3),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M213">View MathML</a>

Hence when R is large enough, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M214">View MathML</a>

Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M199">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M217">View MathML</a>. Therefore we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M218">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M219">View MathML</a>

(3.12)

By (3.9)-(3.12), we derive that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M220">View MathML</a>

(3.13)

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M221">View MathML</a>.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M222">View MathML</a> let us consider the measure sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M223">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M224">View MathML</a>

We assume

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M225">View MathML</a>

By the concentration-compactness lemma [24], there exists a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M226">View MathML</a> (denoted in the same way) satisfying one of the three following possibilities.

Compactness: There exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M227">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M86">View MathML</a> there is a radius <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M229">View MathML</a> with the property that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M230">View MathML</a>

Vanishing: For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M229">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M232">View MathML</a>

Dichotomy: There exists a number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M234">View MathML</a>, such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M86">View MathML</a> there is a number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M229">View MathML</a> and a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M237">View MathML</a> with the following property: Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M238">View MathML</a> there are non-negative measures <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M239">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M240">View MathML</a> such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M241">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M242">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M243">View MathML</a>,

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M244">View MathML</a>.

We are going to rule out the last two possibilities so that compactness holds. Our first goal is to show that vanishing cannot occur. Otherwise,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M245">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M246">View MathML</a>, contradicting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M133">View MathML</a>.

Now for the harder part. Let η be a smooth nonincreasing cut-off function, defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M248">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M249">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M250">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M251">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M252">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M253">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M254">View MathML</a>. Also, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M255">View MathML</a>. We define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M256">View MathML</a>

a nondecreasing function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M248">View MathML</a>. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M258">View MathML</a>. We show now that dichotomy does not occur. Otherwise there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M259">View MathML</a> such that for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M260">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M227">View MathML</a> the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M223">View MathML</a> splits into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M239">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M240">View MathML</a> with the following properties:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M265">View MathML</a>

(3.14)

If we denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M266">View MathML</a>

(3.14) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M267">View MathML</a>

(3.15)

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M268">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M269">View MathML</a>

Using Dichotomy (iii), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M270">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M271">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M272">View MathML</a>

Now we observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M273">View MathML</a>, therefore

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M274">View MathML</a>

(3.16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M275">View MathML</a>

(3.17)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M276">View MathML</a>

(3.18)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M277">View MathML</a>

(3.19)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M278">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114">View MathML</a>.

Recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M280">View MathML</a> (see (2.3)), which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M281">View MathML</a>

(3.20)

Then using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M282">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M283">View MathML</a> in place of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M123">View MathML</a>, respectively, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M285">View MathML</a>

(3.21)

There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M286">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M287">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M288">View MathML</a>

(3.22)

By (f2) and (f3), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M289">View MathML</a>, we see <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M290">View MathML</a> cannot go zero, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M291">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M292">View MathML</a>, by (3.21), (3.22) and (f4), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M293">View MathML</a>

(3.23)

since (f5). By (3.15),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M294">View MathML</a>

(3.24)

a contradiction. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M295">View MathML</a>. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M296">View MathML</a>, we will show <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M297">View MathML</a>. By (3.21) and (3.22), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M298">View MathML</a>

Hence by the Lebesgue dominated convergence theorem, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M299">View MathML</a>

By (f5), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M297">View MathML</a>. Similarly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M301">View MathML</a>. Using this together with (3.16), (3.17), (3.18) and (3.19), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M302">View MathML</a>

Contradiction! Thus dichotomy does not occur.

With vanishing and dichotomy ruled out, we obtain the compactness of a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M223">View MathML</a>, i.e., there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M227">View MathML</a> and for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M86">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M229">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M307">View MathML</a>

(3.25)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M237">View MathML</a> must be bounded, for otherwise (3.25) would imply, in the limit <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M310">View MathML</a>

(3.26)

for some positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M311">View MathML</a>, independent of δ, which implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M312">View MathML</a>, contrary to (3.8).

From the foregoing, it follows that there exist bounded nonnegative measures <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M313">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M314">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M13">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M316">View MathML</a> weakly and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M317">View MathML</a> tightly as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114">View MathML</a>. Lemma I.1 in [27] declares that there exist sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M319">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M320">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M321">View MathML</a>

(3.27)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M322">View MathML</a> denotes a Dirac measure, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M323">View MathML</a>. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M324">View MathML</a> in the support of the singular part of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M313">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M314">View MathML</a>. We consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M327">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M328">View MathML</a>

(3.28)

Choosing the test function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M329">View MathML</a>, from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M330">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M331">View MathML</a>

(3.29)

This reduces to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M332">View MathML</a>

hence (3.27)(3) states

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M333">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M334">View MathML</a>

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M335">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M336">View MathML</a>

(3.30)

which implies that the set J is at most finite. Here CardJ is the cardinal numbers of set J. Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M337">View MathML</a>

(3.31)

since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M338">View MathML</a>

When n is large enough, recall <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M339">View MathML</a> (see (2.11)), together with (3.30) and (3.31), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M340">View MathML</a>

a contradiction. Therefore J is empty, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M341">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114">View MathML</a>. By the Brezis and Lieb lemma [26] again, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M343">View MathML</a>

(3.32)

Equation (3.25) and compact embedding theorem imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M344">View MathML</a>

(3.33)

This together with (3.13), (3.20) and (3.32) allows us to deduce easily

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M345">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M346">View MathML</a> is a uniformly convex Banach space, hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M347">View MathML</a>

(3.34)

From (3.32), (3.33) and (3.34), we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M348">View MathML</a>

(3.35)

i.e., w is the ground state solution of (2.2) in view of (3.13). The proof of Proposition 3.1 is complete. □

4 Proof of Theorem 1.1

Proposition 4.1Supposefsatisfies (f2)-(f4). Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51">View MathML</a>satisfies the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M47">View MathML</a>-condition for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M351">View MathML</a>, that is, every sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M129">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M63">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M354">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M355">View MathML</a>, as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114">View MathML</a>, possesses a convergent subsequence.

Proof Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M129">View MathML</a> is a sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M63">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M359">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M355">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114">View MathML</a>. Using (f4), by a change of variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M362">View MathML</a>, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M363">View MathML</a>

(4.1)

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M129">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M365">View MathML</a>. Therefore we may assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M366">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M365">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M368">View MathML</a> a.e. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M369">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M370">View MathML</a>

and by the Brezis-Lieb lemma [26],

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M371">View MathML</a>

For convenience, we denote by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M372">View MathML</a>

(4.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M373">View MathML</a>

(4.3)

It is clear that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M374">View MathML</a>

(4.4)

It is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M375">View MathML</a>. Hence we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M376">View MathML</a>

and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M377">View MathML</a>

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M378">View MathML</a> by (f2) and (f3). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M379">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M380">View MathML</a>, hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M381">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114">View MathML</a>, and we can obtain the desired conclusion. Hence it remains to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M383">View MathML</a>. By a change of variable, from

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M384">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M385">View MathML</a>

we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M386">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M387">View MathML</a>

(4.5)

By the Sobolev inequalities,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M388">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M390">View MathML</a>, so either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M391">View MathML</a> which contradicts (4.5) or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M379">View MathML</a>. □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M393">View MathML</a> be fixed. Let η be a smooth nonincreasing cut-off function, defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M248">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M249">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M396">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M251">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M398">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M253">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M400">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M85">View MathML</a>. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M118">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M403">View MathML</a>

where w is the positive ground state of (2.2). We may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M404">View MathML</a> is the unique positive number such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M405">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M406">View MathML</a> be any positive function tending to 0 as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M407">View MathML</a>, we define the sublevel

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M408">View MathML</a>

(4.6)

By Lemma 4.2 below, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M48">View MathML</a> is not empty for ε sufficiently small. By noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M410">View MathML</a>, we can define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M411">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M412">View MathML</a>

Lemma 4.2Uniformly in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M118">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M414">View MathML</a>

(4.7)

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M118">View MathML</a>. Computing directly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M416">View MathML</a>

(4.8)

By a change of variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M417">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M418','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M418">View MathML</a>

(4.9)

uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M118">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M420">View MathML</a>

(4.10)

By the exponential decay of ω, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M421">View MathML</a>

(4.11)

uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M118">View MathML</a>. Therefore, in the limit that ε is very small, thanks to (4.8) (4.9) and (4.11), we find

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M423">View MathML</a>

(4.12)

On the other hand, following the idea of [21,25], from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M424">View MathML</a>, by the change of variables <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M425">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M426">View MathML</a>

(4.13)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M427">View MathML</a>

(4.14)

which contradicts (4.12). Thus, up to a subsequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M428">View MathML</a>.

Since f has subcritical growth and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M410">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M170">View MathML</a>. Thus, we can take the limit in (4.13) to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M431">View MathML</a>

(4.15)

from which it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M432">View MathML</a>. Since w also belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M433','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M433">View MathML</a>, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M434">View MathML</a>. This and Lebesgue’s theorem imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M435">View MathML</a>

(4.16)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M436','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M436">View MathML</a>

(4.17)

uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M118">View MathML</a>. Noting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M438">View MathML</a>, from (4.12), (4.16) and (4.17), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M439">View MathML</a>

Thus (4.7) is proved. □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M440">View MathML</a> be the center of mass of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M441">View MathML</a> in terms of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M442','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M442">View MathML</a> norm:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M443">View MathML</a>

Lemma 4.3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M113">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114">View MathML</a>. Then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M446">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M447">View MathML</a>

uniformly for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M118">View MathML</a>.

Proof By change of variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M127">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M450">View MathML</a>

By Proposition 3.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M451','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M451">View MathML</a> converges strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M60">View MathML</a> to w, which is a positive ground state solution of equation (2.2). Thanks to the exponential decay of w (see (2.4)), we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M453','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M453">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M114">View MathML</a>. This completes the proof of Lemma 4.3. □

Proof of Theorem 1.1 By Proposition 4.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51">View MathML</a> satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M47">View MathML</a>-condition for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M457">View MathML</a>. Now let us choose a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M458">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M459">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M407">View MathML</a> and such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M461','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M461">View MathML</a> is not a critical level for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51">View MathML</a>. For such <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M406">View MathML</a>, let us introduce the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M464">View MathML</a> as in (4.6). Then the standard Ljusternik-Schnirelmann theory implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51">View MathML</a> has at least <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M50">View MathML</a> critical points on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M48">View MathML</a> (also see [21]).

By Lemma 4.3, we can assume that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M86">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M469">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M470','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M470">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M471','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M471">View MathML</a>. For such ε, by Lemma 4.2, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M472">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M118">View MathML</a>, thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M474','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M474">View MathML</a>. Recall <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M438">View MathML</a>, calculating directly, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M476','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M476">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M407">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M118">View MathML</a>. Hence the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M479">View MathML</a> is homotopical equivalence to the inclusion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M480">View MathML</a> for ε small enough. We denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M481','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M481">View MathML</a>. It is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M482','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M482">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M483','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M483">View MathML</a> (cf. [[19], Lemma 2.2]). Hence we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M484','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M484">View MathML</a>

Next we show that if u is a critical point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M57">View MathML</a>, then u cannot change sign. Indeed, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M487','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M487">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M488','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M488">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M489','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M489">View MathML</a>, then from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M490','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M490">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M491','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M491">View MathML</a>

By change of variable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M127">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M493','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M493">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M494','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M494">View MathML</a>

Also, noting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M495','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M495">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M496','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M496">View MathML</a>

which is a contradiction. Therefore there exist at least <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M58">View MathML</a> nonzero critical points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M51">View MathML</a> and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M58">View MathML</a> solutions of equation (1.4). □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

WL carried out the genetic studies, participated in the sequence alignment and drafted the manuscript. PZ checked the references.

Acknowledgements

This work was partially supported by the National Natural Science Foundation of China (11261052). The authors are grateful to Prof. Guowei Dai for pointing out several mistakes and valuable comments.

References

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

  2. Oh, Y-G: Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M500','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M500">View MathML</a>. Commun. Partial Differ. Equ.. 13, 1499–1519 (1988). Publisher Full Text OpenURL

  3. Oh, Y-G: Corrections to ‘Existence of semi-classical bound state of nonlinear Schrödinger equations with potentials of the class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M502','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/199/mathml/M502">View MathML</a>’. Commun. Partial Differ. Equ.. 14, 833–834 (1989)

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

  5. Del Pino, M, Felmer, P: Local mountain passes for semilinear elliptic problems in unbounded domains. Calc. Var.. 4, 121–137 (1996). Publisher Full Text OpenURL

  6. Del Pino, M, Felmer, P: Spike-layered solutions of singularly perturbed elliptic problems in a degenerate setting. Indiana Univ. Math. J.. 48(3), 883–898 (1999)

  7. Rabinowitz, P: On a class of nonlinear Schrödinger equations. Z. Angew. Math. Phys.. 43, 270–291 (1992). Publisher Full Text OpenURL

  8. Wang, X: On concentration of positive bound states of nonlinear Schrödinger equations. Commun. Math. Phys.. 153, 229–244 (1993). Publisher Full Text OpenURL

  9. Wang, X, Zeng, B: On concentration of positive bound states of nonlinear Schrödinger equation with competing potential functions. SIAM J. Math. Anal.. 28, 633–655 (1997). Publisher Full Text OpenURL

  10. Del Pino, M, Felmer, P: Multi-peak bound states of nonlinear Schrödinger equations. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 15(2), 127–149 (1998). Publisher Full Text OpenURL

  11. Gui, C: Existence of multi-bump solutions for nonlinear Schrödinger equations. Commun. Partial Differ. Equ.. 21, 787–820 (1996). Publisher Full Text OpenURL

  12. Lin, C, Ni, W-M, Takagi, I: Large amplitude stationary solutions to a chemotaxis systems. J. Differ. Equ.. 72, 1–27 (1988). Publisher Full Text OpenURL

  13. Ni, WM, Takagi, I: On the shape of least-energy solutions to a semilinear Neumann problem. Commun. Pure Appl. Math.. 45, 819–851 (1990)

  14. Ni, WM, Takagi, I: Locating the peaks of least-energy solutions to a semilinear Neumann problem. Duke Math. J.. 70, 247–281 (1993). Publisher Full Text OpenURL

  15. Wang, ZQ: On the existence of multiple, single peaked solutions for a semilinear Neumann problem. Arch. Ration. Mech. Anal.. 120, 375–399 (1992). Publisher Full Text OpenURL

  16. Gui, C: Multi-peak solutions for a semilinear Neumann problem. Duke Math. J.. 84, 739–769 (1996). Publisher Full Text OpenURL

  17. Li, YY: On a singularly perturbed equation with Neumann boundary condition. Commun. Partial Differ. Equ.. 23, 487–545 (1998)

  18. Wei, J, Winter, M: Multiple boundary spike solutions for a wide class of singular perturbation problems. J. Lond. Math. Soc.. 59(2), 585–606 (1999). 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. Ambrosetti, A, Malchiodi, A: Nonlinear Analysis and Semilinear Elliptic Problems, Cambridge University Press, Cambridge (2007)

  21. Liu, W, Zhao, P: Critical semilinear Neumann problem with magnetic fields. Preprint

  22. Berestycki, H, Lions, PL: Nonlinear scalar field equations I. Existence of a ground state. Arch. Ration. Mech. Anal.. 82, 313–345 (1983)

  23. Zhang, J, Zou, W: A Berestycki-Lions theorem revisited. Commun. Contemp. Math.. 14(5), (2012) Article ID 1250033

  24. Lions, PL: The concentration-compactness principle in the calculus of variation. The locally compact case. II. Ann. Inst. Henri Poincaré, Anal. Non Linéaire. 1, 223–283 (1984)

  25. Alves, CO, Figueiredo, GM, Furtado, MF: Multiple solutions for a nonlinear Schrödinger equation with magnetic fields. Commun. Partial Differ. Equ.. 36, 1565–1586 (2011). Publisher Full Text OpenURL

  26. Brezis, H, Lieb, E: A relation between pointwise convergence of functions and convergence of functional. Proc. Am. Math. Soc.. 88, 486–490 (1983)

  27. Lions, PL: The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoam.. 1, 145–201 (1985)