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Existence of solutions for semilinear elliptic equations on R N

Ruichang Pei12* and Jihui Zhang1

Author Affiliations

1 Institute of Mathematics, School of Mathematics and Computer Sciences, Nanjing Normal University, Nanjing, 210097, P.R. China

2 School of Mathematics and Statistics, Tianshui Normal University, Tianshui, 741001, P.R. China

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Boundary Value Problems 2013, 2013:163  doi:10.1186/1687-2770-2013-163


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/163


Received:28 March 2013
Accepted:8 June 2013
Published:9 July 2013

© 2013 Pei and Zhang; licensee Springer

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

Abstract

In this paper, the existence of at least one nontrivial solution for a class of semilinear elliptic equations on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M1">View MathML</a> is established by using the linking methods.

Keywords:
Schrödinger equation; subcritical exponent; local linking

1 Introduction

In this paper we consider the question of the existence of solutions for a class of semilinear equations of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M3">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M4">View MathML</a> is a parameter and the nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M5">View MathML</a> is asymptotically linear, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M6">View MathML</a>

(1.1)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M8">View MathML</a>. In case this equation is considered in a bounded domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M9">View MathML</a> (with, say, the Dirichlet boundary condition), there is a large amount of literature on existence and multiplicity results, with the case of resonance being of particular interest (see [1-3]). We recall that the problem is said to be at resonance if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M10">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M11">View MathML</a> denotes the spectrum of S, the ‘asymptotic linearization’ of the problem. In other words, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M12">View MathML</a> is the operator given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M13">View MathML</a>

(1.2)

On the other hand, a systematic study of such asymptotically linear problems set in unbounded domains or the whole space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M14">View MathML</a> is more recent and presents a number of mathematical difficulties (see [4,5]). As an example, we note that in the case of problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M15">View MathML</a>), the asymptotic linearization operator S (now defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M16">View MathML</a>) has a much more complicated spectrum (including an essential part <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M17">View MathML</a>), which in turn makes the study of this problem more challenging. In [4], motivated by the paper [5], Tehrani and Costa studied the existence of positive solutions to (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M15">View MathML</a>) by using the mountain pass theorem if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M19">View MathML</a> satisfies some strong asymptotically linear conditions. Comparing with previous paper [4], in [6], Tehrani obtained the existence of a (possibly sign-changing) solution for problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M15">View MathML</a>) under essentially condition (1.1) only. In fact, he proved the following.

Theorem 1.0[6]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M21">View MathML</a>and assume that

(G) for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M22">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M23">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M24">View MathML</a>

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M25">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M26">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M27">View MathML</a>, then (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M15">View MathML</a>) has a solution in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M29">View MathML</a>.

Now, one naturally asks: Are there nontrivial solutions for problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M15">View MathML</a>) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M10">View MathML</a> in the above theorem? Obviously, this case is resonance. But, this problem is not easy because we face the difficulties of verifying that the energy functional satisfies the (PS) condition if we still follow the idea of [6]. Here, there is still an interesting problem: Are there nontrivial solutions for problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M15">View MathML</a>) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M10">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M34">View MathML</a> (in Theorem 1.0) is more generalized superlinear? We will answer the above problems affirmatively by using Li and Willem’s local linking methods (see [7]).

Next, we recall a few basic facts in the theory of Schrödinger operators which are relevant to our discussion (see [6]).

1. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M35">View MathML</a>, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M36">View MathML</a>.

2. The bottom of the spectrum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M11">View MathML</a> of the operator S is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M38">View MathML</a>

Therefore we clearly have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M39">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M40">View MathML</a>, then by using the concentration compactness principle of Lions, one shows that Λ is the principle eigenvalue of S with a positive eigenfunction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M41">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M42">View MathML</a>

3. The spectrum of S in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M43">View MathML</a>, namely <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M44">View MathML</a>, is at most a countable set, which we denote by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M45">View MathML</a>

where each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M46">View MathML</a> is an isolated eigenvalue of S of the finite multiplicity. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M47">View MathML</a>denote the eigenspace of S corresponding to the eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M48">View MathML</a>.

Now, we state our main results. In this paper, we always assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M49">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M50">View MathML</a>. The conditions imposed on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M51">View MathML</a> (see Theorem 1.0) are as follows:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M52">View MathML</a>, and there are constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M53">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M54">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M55">View MathML</a>;

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M56">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M57">View MathML</a>, uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M14">View MathML</a>;

(H3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M59">View MathML</a> uniformly on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M14">View MathML</a>;

(H4) There is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M61">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M62">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M63">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M64">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M65">View MathML</a>;

(H5) For some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M66">View MathML</a>, either

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M67">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M68">View MathML</a>

(H6) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M69">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M70">View MathML</a>.

Theorem 1.1Assume that conditions (H1)-(H4) hold. Ifλis an eigenvalue of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M71">View MathML</a>, assume also that (H5) and (H6) hold. Then the problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M15">View MathML</a>) has at least one nontrivial solution.

Remark 1.1 It follows from the condition (H3) that our nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M51">View MathML</a> does not satisfy the classical condition of Ambrosetti and Rabinowitz:

(AR) There is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M74">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M75">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M77">View MathML</a>.

In recent years, there have been some papers devoted to replacing (AR) with more natural conditions (see [8-10]). But our methods are different from the references therein.

We also consider asymptotically quadratic functions. We assume that:

(H7) For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M22">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M79">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M80">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M81">View MathML</a>.

Theorem 1.2Assume that conditions (H2), (H6), (H7) and one of the following conditions hold:

(A1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M82">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M83">View MathML</a>;

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M84">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M83">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M66">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M87">View MathML</a>

(A3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M83">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M66">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M91">View MathML</a>

Then problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M15">View MathML</a>) has at least one nontrivial solution.

2 Preliminaries

Let X be a Banach space with a direct sum decomposition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M93">View MathML</a>

Consider two sequences of subspaces

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M94">View MathML</a>

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M95">View MathML</a>

For every multi-index <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M96">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M97">View MathML</a>. We know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M98">View MathML</a>

A sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M99">View MathML</a> is admissible if, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M100">View MathML</a>, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M101">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M102">View MathML</a>. For every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M103">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M104">View MathML</a> the function I restricted <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M105">View MathML</a>.

Definition 2.1 Let I be locally Lipschitz on X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M106">View MathML</a>. The functional I satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M107">View MathML</a> condition if every sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M108">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M109">View MathML</a> is admissible and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M110">View MathML</a>

contains a subsequence which converges to a critical point of I.

Definition 2.2 Let I be locally Lipschitz on X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M106">View MathML</a>. The functional I satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M112">View MathML</a> condition if every sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M108">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M109">View MathML</a> is admissible and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M115">View MathML</a>

contains a subsequence which converges to a critical point of I.

Remark 2.1 1. The <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M112">View MathML</a> condition implies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M107">View MathML</a> condition for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M106">View MathML</a>.

2. When the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M107">View MathML</a> sequence is bounded, then the sequence is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M120">View MathML</a> sequence (see [11]).

3. Without loss of generality, we assume that the norm in X satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M121">View MathML</a>

Definition 2.3 Let X be a Banach space with a direct sum decomposition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M122">View MathML</a>

The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M123">View MathML</a> has a local linking at 0, with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M124">View MathML</a> if, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M70">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M126">View MathML</a>

Lemma 2.1 (see [7])

Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M123">View MathML</a>satisfies the following assumptions:

(B1) Ihas a local linking at 0 and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M128">View MathML</a>;

(B2) Isatisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M129">View MathML</a>;

(B3) Imaps bounded sets into bounded sets;

(B4) For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M131">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M133">View MathML</a>. ThenIhas at least two critical points.

Remark 2.2 Assume that I satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M107">View MathML</a> condition. Then this theorem still holds.

Let X be a real Hilbert space and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M123">View MathML</a>. The gradient of I has the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M136">View MathML</a>

where A is a bounded self-adjoint operator, 0 is not the essential spectrum of A, and B is a nonlinear compact mapping.

We assume that there exist an orthogonal decomposition,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M137">View MathML</a>

and two sequences of finite-dimensional subspaces,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M138">View MathML</a>

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M139">View MathML</a>

For every multi-index <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M96">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M105">View MathML</a> the space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M142">View MathML</a>

by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M143">View MathML</a> the orthogonal projector onto <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M105">View MathML</a>, and by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M145">View MathML</a> the Morse index of a self-adjoint operator L.

Lemma 2.2 (see [7])

Isatisfies the following assumptions:

(i) Ihas a local linking at 0 with respect to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M124">View MathML</a>;

(ii) There exists a compact self-adjoint operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M147">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M148">View MathML</a>

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M149">View MathML</a>is invertible;

(vi) For infinitely many multiple-indices<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M150">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M151">View MathML</a>

ThenIhas at least two critical points.

3 The proof of main results

Proof of Theorem 1.1 (1) We shall apply Lemma 2.1 to the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M152">View MathML</a>

defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M153">View MathML</a>. We consider only the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M10">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M155">View MathML</a>

(3.1)

Then other case is similar and simple.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M156">View MathML</a> be a finite dimensional space spanned by the eigenfunctions corresponding to negative eigenvalues of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M157">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M158">View MathML</a> be its orthogonal complement in X. Choose a Hilbertian basis <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M159">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M160">View MathML</a>) for X and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M161">View MathML</a>

By the condition (H1) and Sobolev inequalities, it is easy to see that the functional I belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M162">View MathML</a> and maps bounded sets to bounded sets.

(2) We claim that I has a local linking at 0 with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M124">View MathML</a>. Decompose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M158">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M165">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M166">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M167">View MathML</a>. Also, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M169">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M170">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M171">View MathML</a>.

For the convenience of our proof, we state some facts for the norm of the whole space X. It is well known that there is an equivalent norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M172">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M153">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M174">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M175">View MathML</a>

By the equivalence of norm in the finite-dimensional space, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M176">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M177">View MathML</a>

(3.2)

It follows from (H1) and (H2) that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M22">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M179">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M180">View MathML</a>

(3.3)

Hence, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M181">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M182">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M183">View MathML</a> is a constant and hence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M70">View MathML</a> small enough,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M185">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M186">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M187">View MathML</a> and let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M188">View MathML</a>

From (3.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M189">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M190">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M191">View MathML</a>. On the one hand, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M192">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M193">View MathML</a>. Hence, from (H5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M194">View MathML</a>

On the other hand, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M195">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M196">View MathML</a>. It follows from (3.3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M197">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M196">View MathML</a> and all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M199">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M190">View MathML</a>, which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M201">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M202">View MathML</a> is a constant. Hence, there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M203">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M204">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M205">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M206">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M169">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M190">View MathML</a>, which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M209">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M210">View MathML</a> small enough.

(3) We claim that I satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M211">View MathML</a>. Consider a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M108">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M108">View MathML</a> is admissible and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M214">View MathML</a>

(3.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M215">View MathML</a>

(3.5)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M216">View MathML</a>. Up to a subsequence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M217">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M218">View MathML</a>, we choose a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M219">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M220">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M182">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M222">View MathML</a>. Now, we claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M223">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M22">View MathML</a>; for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M225">View MathML</a>, then,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M226">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M227">View MathML</a>, we may fix r large enough such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M228">View MathML</a>

for all n. Moreover, by (H6), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M229">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M230">View MathML</a>

for all n. Finally, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M231">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M232">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M233">View MathML</a>, we can use (H1) again to derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M234">View MathML</a>

for n large enough. Combining the above three formulas, our claim holds.

So, for n large enough, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M235">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M236">View MathML</a>

(3.6)

where ϵ is a small enough constant.

That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M237">View MathML</a>. Now, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M238">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M239">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M240">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M241">View MathML</a>

(3.7)

Therefore, using (H4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M242">View MathML</a>

This contradicts (3.5).

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M243">View MathML</a>, then the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M244">View MathML</a> has a positive Lebesgue measure. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M245">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M246">View MathML</a>. Hence, by (H3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M247">View MathML</a>

(3.8)

From (3.4), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M248">View MathML</a>

(3.9)

By (3.8), the right-hand side of (3.9) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M249">View MathML</a>. This is a contradiction.

In any case, we obtain a contradiction. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M250">View MathML</a> is bounded.

Next, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M250">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M252">View MathML</a> and prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M252">View MathML</a> contains a convergent subsequence.

In fact, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M252">View MathML</a> is bounded in X. Passing to a subsequence, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M255">View MathML</a> in X. In order to establish strong convergence, it suffices to show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M256">View MathML</a>

By the condition (H6) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M257">View MathML</a>, we can similarly conclude it according to the above proof of our claim.

Finally, we claim that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M130">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M259">View MathML</a>

By (H2) and (H3), there exist large enough M and some positive constant T such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M260">View MathML</a>

So, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M261">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/163/mathml/M262">View MathML</a>

Hence, our claim holds. □

Proof of Theorem 1.2 We omit the proof which depends on Lemma 2.2 and is similar to the preceding one since our result is a variant of Ding Yanheng’s Theorem 1.2 (see [12]). □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors read and approved the final manuscript.

Acknowledgements

The authors would like to thank the referees for valuable comments and suggestions in improving this paper. This work was supported by the National NSF (Grant No. 10671156) of China.

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