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The existence of positive solutions to an elliptic system with nonlinear boundary conditions

Chunhua Wang* and Jing Yang

Author Affiliations

School of Mathematics and Statistics, Huazhong Normal University, Wuhan, 430079, P.R. China

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Boundary Value Problems 2013, 2013:159  doi:10.1186/1687-2770-2013-159


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/159


Received:8 April 2013
Accepted:16 June 2013
Published:1 July 2013

© 2013 Wang and Yang; licensee Springer

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

Abstract

In this paper, we consider the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M1">View MathML</a>

where Ω is a bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M2">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M3">View MathML</a>) with smooth boundary, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M4">View MathML</a> is the outer normal derivative and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M5">View MathML</a> are positive and continuous functions. Under certain assumptions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M7">View MathML</a>, but without the usual (AR) condition, we prove that the problem has at least one positive strong pair solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M8">View MathML</a> (see Definition 1.4 below) by applying a linking theorem for strong indefinite functional.

MSC: 35A01, 35J20, 35J25.

Keywords:
fractional Sobolev spaces; linking theorem; nonlinear boundary conditions; strong solution

1 Introduction and main result

In this paper, we mainly study the following system:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M9">View MathML</a>

(1.1)

where Ω is a bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M2">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M3">View MathML</a>) with smooth boundary, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M4">View MathML</a> is the outer normal derivative and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M5">View MathML</a> are positive and continuous functions.

Existence results for nonlinear elliptic systems have received a lot of interest in recent years (see [1-12]), particularly when the nonlinear term appears as a source in the equation, complemented with Dirichlet boundary conditions. To our knowledge, about the system with nonlinear boundary conditions, there are not many results. Here we refer to [9,13,14].

We are mainly motivated by [12] and [14].

In [12], Li and one of the authors considered

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M14">View MathML</a>

(1.2)

Under some given conditions, we proved that (1.2) had at least one positive solution pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M15">View MathML</a>.

In [14], Bonder, Pinasco, Rossi studied

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M16">View MathML</a>

(1.3)

They assumed that H satisfied the following conditions:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M17">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M18">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M19">View MathML</a>) The Ambrosetti-Rabinowitz type condition: For R large, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M20">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M21">View MathML</a>

(1.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M23">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M24">View MathML</a>

(1.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M25">View MathML</a>

(1.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M26">View MathML</a>

(1.7)

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M27">View MathML</a>, they also assumed

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M28">View MathML</a>

(1.8)

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M29">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M31">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M32">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M33">View MathML</a>.

They obtained infinitely many nontrivial solutions of (1.3) under the assumptions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M17">View MathML</a>) to (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M32">View MathML</a>) by using variational arguments and a fountain theorem. Note that (1.4) implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M36">View MathML</a>

(1.9)

(see Lemma 1.1 in [6]). Therefore it is not difficult to verify that any (PS) sequence (or (C)c sequence) of the corresponding functional is bounded in some suitable space.

The crucial part in the nonlinear boundary conditions case is to find the proper functional setting for (1.1) that allows us to treat our problem variationally. We accomplish this by defining a self-adjoint operator that takes into account the boundary conditions together with the equations and considering its fractional powers that satisfy a suitable ‘integration by parts’ formula. In order to obtain nontrivial solutions, we use a linking theorem (see [11]).

The assumptions we impose on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M7">View MathML</a> are as follows:

(H1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M39">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M40">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M41">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M42">View MathML</a>

(H2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M43">View MathML</a> uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M44">View MathML</a>.

(H3) There is a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M45">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M46">View MathML</a>

(1.10)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M47">View MathML</a> and satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M48">View MathML</a>

(1.11)

(H4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M49">View MathML</a> uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M44">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M52">View MathML</a>.

(H5) For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M44">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M56">View MathML</a>, there are two positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M58">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M59">View MathML</a>

(1.12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M60">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M61">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M62">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M44">View MathML</a>.

Remark 1.1 By (1.11), there exist l and m with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M66">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M67">View MathML</a>

(1.13)

Remark 1.2 (H5) was first introduced by Miyagaki and Souto in [15]. A typical pair of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M47">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M73">View MathML</a> satisfy (H1) to (H5). However, the pair of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M63">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M73">View MathML</a> satisfy (H5) but do not satisfy the usual (AR) condition and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M19">View MathML</a>) in this paper.

Remark 1.3 The assumptions we impose on f and g are different from the assumptions in [14]. To our best knowledge, it is the first time the group assumptions have been used to deal with a system with nonlinear boundary conditions.

In order to state our main result, first we give a definition.

Definition 1.4 We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M8">View MathML</a> is a strong solution of (1.1) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M81">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M8">View MathML</a> satisfies (1.1) a.e. in Ω.

Our main results is as follows.

Theorem 1.5Let (H1)-(H5) hold. Then system (1.1) possesses at least one positive strong solution pair<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M83">View MathML</a>.

The main difficulties to deal with system (1.1) consist in at least three aspects. Firstly, due to the type of growth of the functions f and g, we cannot work with the usual <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M84">View MathML</a>, and then we need fractional Sobolev spaces. Secondly, although we have a variational problem, the functional associated to it always has a strong indefinite quadratic part. So, the functional possesses no mountain-pass structure but the linking geometric structure, which is more complicated to handle. Thirdly, as we do not assume that the functions f and g satisfy the (AR) conditions, it is much more difficult to show that any (C)c sequence is uniformly bounded in E (see Section 2).

To prove Theorem 1.5, we try to find a critical point of the functional Φ (see (2.5)) in E. We prove that Φ has a linking geometric structure and use a linking theorem under (C)c condition (see Theorem 2.1 in [11]) to get a (C)c-sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M85">View MathML</a> of Φ. The main difficulty now will be to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M86">View MathML</a> is uniformly bounded in E without the (AR) condition. Then we prove that any (C)c-sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M85">View MathML</a> of Φ is bounded. To overcome this difficulty, we use some techniques used in [12,16] for which the assumptions (H4), (H5) play important roles. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M86">View MathML</a> is bounded, then we can prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M86">View MathML</a> has a subsequence which converges to a nontrivial critical point of Φ. Hence, by the strong maximum principle, we can prove that the pair solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M8">View MathML</a> is positive.

The paper is organized as follows. In Section 2, we give some preliminaries. We prove our main result in Section 3.

2 Some preliminaries

In this section we mainly give some preliminaries which will be used in Section 3. We follow the structure in [13].

Throughout this paper, we consider the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M91">View MathML</a> which is a Hilbert space with the inner product, which we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M92">View MathML</a>, given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M93">View MathML</a>

Now we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M94">View MathML</a> be the operator defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M95">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M96">View MathML</a>. It is not difficult to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M97">View MathML</a> is dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M98">View MathML</a>. Note that A is invertible with its inverse given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M99">View MathML</a>

where u is the solution of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M100">View MathML</a>

(2.1)

By standard regularity (see [17]), it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M101">View MathML</a> is bounded and compact. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M102">View MathML</a>. Therefore, in order to see that A (hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M101">View MathML</a>) is self-adjoint, it suffices to prove that A is symmetric ([18], p.512). In fact, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M104">View MathML</a>, applying Green’s formula, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M105">View MathML</a>

Hence A is symmetric. Also we can check that A (and so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M101">View MathML</a>) is positive. For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M107">View MathML</a> and by Green’s formula again, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M108">View MathML</a>

Hence there is a sequence of eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M109">View MathML</a> with eigenfunctions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M110">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M111">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M112">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M113">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M114">View MathML</a>

(2.2)

Now we consider the following fractional powers of A, i.e., for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M115">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M116">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M117">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M118">View MathML</a>, which is a Hilbert space under the inner product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M119">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M120">View MathML</a>. Indeed, if we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M121">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M122">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M123">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M124">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M125">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M126">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M127">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M128">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M129">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M130">View MathML</a>

Noting that Ω is smooth, it follows from the results of p.187 in [19] (see also [18,20]) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M131">View MathML</a>

The following compact result will be useful later.

Proposition 2.1 (Theorem 2.1, [14])

Given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M132">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M133">View MathML</a>so that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M134">View MathML</a>, the inclusion map<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M135">View MathML</a>is well defined and bounded. Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M136">View MathML</a>, then the inclusion is compact.

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M137">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M65">View MathML</a>, l, m are the same as in Remark 1.1 and define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M139">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M140">View MathML</a>

Associated to B, we have the quadratic form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M141">View MathML</a>

It is easy to see (one can refer to [14]) that the bounded self-adjoint operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M142">View MathML</a> defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M143">View MathML</a> has exactly two eigenvalues +1 and −1, and that the corresponding eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M144">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M145">View MathML</a> are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M146">View MathML</a>

where we use the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M147">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M148">View MathML</a>. The spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M144">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M145">View MathML</a> are orthogonal with respect to the bilinear B, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M151">View MathML</a>

Moreover, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M152">View MathML</a>

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M153">View MathML</a>. We see also that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M155">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M156">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M157">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M158">View MathML</a>

(2.3)

From (1.10), Remark 1.1 and Proposition 2.1, we can define the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M159">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M160">View MathML</a>

(2.4)

Lemma 2.2The functionaldefined by (2.4) is of class<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M161">View MathML</a>and its derivative is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M162">View MathML</a>

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M163">View MathML</a>is compact.

Proof From (1.10), Hölder’s inequality and Proposition 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M164">View MathML</a>

Similarly, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M165">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M163">View MathML</a> is well defined and bounded in E. A standard argument yields that ℋ is Fréchet differentiable with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M163">View MathML</a> continuous. By Proposition 2.1 we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M163">View MathML</a> is compact (see [21] for the details). □

Now we define the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M169">View MathML</a> for (1.1) given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M170">View MathML</a>

(2.5)

Moreover, Φ is class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M171">View MathML</a>.

Definition 2.3 We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M172">View MathML</a> is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M173">View MathML</a>-weak solution of (1.1) if z is a critical point of Φ. In other words, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M174">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M175">View MathML</a>

(2.6)

Now we give a regularity result of an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M173">View MathML</a> weak solution.

Proposition 2.4If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M177">View MathML</a>is an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M173">View MathML</a>-weak solution of (1.1), then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M179">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M180">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M181">View MathML</a>

(2.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M182">View MathML</a>

(2.8)

In other words, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M8">View MathML</a>is a strong solution of (1.1).

Proof Although the proof is only needed to make some minor modifications as that of Theorem 2.2 in [13], for the readers’ convenience, we give its detailed proof.

Let us consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M184">View MathML</a> in (2.6), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M185">View MathML</a>

(2.9)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M186">View MathML</a>.

If we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M187">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M188">View MathML</a>

(2.10)

On the other hand, by (1.10) and Proposition 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M189">View MathML</a>

(2.11)

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M190">View MathML</a>. Then from basic elliptic theory (Theorem 9.9, p.9, [17]) there exists one function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M191">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M192">View MathML</a>

Then we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M193">View MathML</a>

(2.12)

From (2.10), (2.11) and (2.12), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M194">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M195">View MathML</a>. We have gotten that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M196">View MathML</a>. Finally, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M195">View MathML</a>, we conclude that u satisfies (2.7). We can make the same argument for v. □

3 The proof of our main result

In this section, we mainly want to prove Theorem 1.5. First we present a linking theorem from [11]. Then we prove that it can be applied to our functional setting stated in Section 2.

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M7">View MathML</a> satisfy the assumptions (H1)-(H3), then it is easy to see that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M200">View MathML</a> there is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M201">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M202">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M203">View MathML</a>

(3.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M204">View MathML</a>

(3.2)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M40">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M207">View MathML</a> is a solution of (1.1). So we are interested in nontrivial and nonnegative solutions of (1.1).

Recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M85">View MathML</a> is called a Palais-Smale sequence of a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M161">View MathML</a> functional I on E at level c ((PS)c-sequence for short) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M210">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M211">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M212">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M213">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M210">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M215">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M212">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M213">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M86">View MathML</a> will be called a Cerami sequence at level c ((C)c-sequence for short). A standard way to prove the existence of a positive solution to (1.1) is to get a (PS)c or (C)c sequence for Φ and then to prove that the sequence converges to a solution to (1.1). In this paper, we want to get a (C)c sequence by a linking theorem (Theorem 2.1, in [11]). So, we need to recall some terminology (see, e.g., [11,22]).

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M219">View MathML</a> be a closed separable subspace of a Hilbert space H with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M220">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M221">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M222">View MathML</a>, we shall write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M223">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M224">View MathML</a>. On H we define a new norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M225">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M226">View MathML</a> is a total orthonormal sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M219">View MathML</a>. The topology generated by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M228">View MathML</a> will be called the τ-topology. Recall from [22] that a homotopy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M229">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M230">View MathML</a>, is called admissible if:

(i) h is τ-continuous, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M231">View MathML</a> in τ-topology as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M213">View MathML</a> whenever <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M233">View MathML</a> in τ-topology and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M234">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M213">View MathML</a>.

(ii) g is τ-locally finite-dimensional, i.e., for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M236">View MathML</a>, there is a neighborhood U of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M237">View MathML</a> in the product topology of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M238">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M239">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M240">View MathML</a> is contained in a finite-dimensional subspace of H.

Admissible maps are defined similarly. Recall also that admissible maps and homotopies are necessarily continuous, and on bounded subsets of H the τ-topology coincides with the product topology of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M241">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M242">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M243">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M244">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M245">View MathML</a> and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M246">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M247">View MathML</a>

Proposition 3.1 (Theorem 2.1, [11])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M248">View MathML</a>be a separable Hilbert space with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M219">View MathML</a>orthogonal to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M250">View MathML</a>. Suppose that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M251">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M252">View MathML</a>is bounded below, weakly sequentially lower semi-continuous and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M253">View MathML</a>is weakly sequentially continuous.

(ii) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M254">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M255">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M244">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M257">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M258">View MathML</a>.

Then there exists a (C)c-sequence for Φ, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M259">View MathML</a>

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M260">View MathML</a>.

For fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M261">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M244">View MathML</a>, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M263">View MathML</a>

Lemma 3.2There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M264">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M255">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M266">View MathML</a>.

Proof For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M267">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M268">View MathML</a>, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M269">View MathML</a> or, equivalently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M270">View MathML</a>. By (3.2) and Proposition 2.1, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M271">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M47">View MathML</a>, we have if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M273">View MathML</a> is small enough,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M274">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M255">View MathML</a>. □

Lemma 3.3For thergiven by Lemma 3.2 and any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M276">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M277">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M278">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M279">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M280">View MathML</a>.

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M281">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M282">View MathML</a> with either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M283">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M284">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M285">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M286">View MathML</a>.

(i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M286">View MathML</a>, then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M288">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M289">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M290">View MathML</a>

since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M291">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M202">View MathML</a>.

(ii) Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M293">View MathML</a>. We argue by contradiction. Suppose that there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M294">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M295">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M296">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M277">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M298">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M299">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M300">View MathML</a>, then by the definitions of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M144">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M145">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M303">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M304">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M305">View MathML</a>

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M306">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M307">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M308">View MathML</a>

(3.3)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M291">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M202">View MathML</a>, by (3.3) we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M311">View MathML</a>.

On the other hand, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M312">View MathML</a>, which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M313">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M314">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M315">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M213">View MathML</a>, where ⇀ denotes the weak convergence in E.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M317">View MathML</a>, then from (3.3) we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M318">View MathML</a>

Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M319">View MathML</a>

which is impossible.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M320">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M321">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M322">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M213">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M324">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M44">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M326">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M327">View MathML</a>

thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M328">View MathML</a>

(3.4)

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M213">View MathML</a>.

Similarly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M330">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M331">View MathML</a>

(3.5)

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M213">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M333">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M291">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M335">View MathML</a>

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M336">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M337">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M338">View MathML</a>. Hence, by Proposition 2.1 we may assume, passing to a subsequence, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M339">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M338">View MathML</a>. By (3.8), (3.9) and (H4), taking limit in (3.10), using Fatou’s lemma and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M341">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M342">View MathML</a>

which is impossible, thus the lemma is proved. □

Lemma 3.4If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M86">View MathML</a>is a (C)c-sequence of Φ, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M86">View MathML</a>is bounded inE.

Proof Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M345">View MathML</a> is a (C)c sequence for Φ, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M346">View MathML</a>

which shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M347">View MathML</a>

(3.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M348">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M349">View MathML</a>.

We suppose, by contradiction, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M350">View MathML</a>

(3.7)

and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M351">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M352">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M353">View MathML</a>

By Proposition 2.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M354">View MathML</a> contains a subsequence, denoted again by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M355">View MathML</a> such that we may assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M356">View MathML</a>

(3.8)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M357">View MathML</a>. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M358">View MathML</a>

and (3.7) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M359">View MathML</a>

We may assume, without loss of generality, that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M360">View MathML</a>

(3.9)

By (H4), we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M361">View MathML</a>

This means that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M362">View MathML</a>

(3.10)

By (H4), there is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M363">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M364">View MathML</a>

(3.11)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M365">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M366">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M367">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M368">View MathML</a> is continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M369">View MathML</a>, there is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M370">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M371">View MathML</a>

(3.12)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M372">View MathML</a>. From (3.11) and (3.12), we see that there is a constant C, such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M373">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M374">View MathML</a>

which shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M375">View MathML</a>

This means that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M376">View MathML</a>

(3.13)

Since by (3.6) we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M377">View MathML</a>

which shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M378">View MathML</a>

(3.14)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M379">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M380">View MathML</a>

(3.15)

We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M381">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M382">View MathML</a>, then by Fatou’s lemma, (H4) and Hölder’s inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M383">View MathML</a>

which is impossible.

This shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M384">View MathML</a>

Hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M385">View MathML</a> a.e. in Ω.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M386">View MathML</a> is continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M387">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M388">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M389">View MathML</a>), such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M390">View MathML</a>

As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M391">View MathML</a>, we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M392">View MathML</a>

By (H5), then we get for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M387">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M394">View MathML</a>

(3.16)

On the other hand, taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M395">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M396">View MathML</a>, by (3.8) then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M397">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M398">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M399">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M400">View MathML</a>). From (3.2) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M398">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M402">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M403">View MathML</a>) as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M213">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M405','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M405">View MathML</a>

So we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M406">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M407','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M407">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M408">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M409">View MathML</a>, we get a contradiction. This proves that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M410">View MathML</a> for some constant C. □

Proof of Theorem 1.5 Under the assumptions (H1)-(H5), we know that the functional Φ given by (2.5) is in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M411">View MathML</a>. By Lemma 3.2, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M264">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M255">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M266">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M415">View MathML</a>. By Lemma 3.3, for such an r, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M278">View MathML</a> and suitable <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M261">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M279">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M419">View MathML</a> was given before Lemma 3.3. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M148">View MathML</a> and for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M421">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M422">View MathML</a>

Since from Proposition 2.1 and Remark 1.1 we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M423','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M423">View MathML</a>, from (3.2) and Fatou’s lemma, we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M424','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M424">View MathML</a>

is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M161">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M426">View MathML</a> is weakly sequentially lower semicontinuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M163">View MathML</a> is weakly sequentially continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M212">View MathML</a>. Hence by Proposition 3.1 there exists a (C)c-sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M86">View MathML</a> for Φ, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M430">View MathML</a>. By Lemma 3.4, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M86">View MathML</a> is bounded in E. So, up to a subsequence, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M432','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M432">View MathML</a> in E, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M213">View MathML</a>. From Lemma 2.2, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M434">View MathML</a> is compact. So it is easy to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M435">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M212">View MathML</a>. Hence z is a nontrivial solution pair of (1.1). Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M421','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M421">View MathML</a> is a nonnegative solution pair of (1.1). Applying the strong maximum principle, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M438','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M438">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M439','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/159/mathml/M439">View MathML</a>. This completes the proof. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

JY is mainly in charge of Section 2 and CW is mainly responsible for Section 3. This paper is finished under of our joint efforts. We discuss many times and make many modifications. Both authors read and approved the final manuscript.

Acknowledgements

The authors were partially supported by NSFC (No. 11071092; No. 11071095; No. 11101171), the PhD specialized grant of the Ministry of Education of China (20110144110001).

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