Open Access Research

Existence of solutions for quasilinear elliptic equations with superlinear nonlinearities

Jia Gao*, Huang Lina and Zhang Xiaojuan

Author Affiliations

College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

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Boundary Value Problems 2012, 2012:90  doi:10.1186/1687-2770-2012-90


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/90


Received:23 March 2012
Accepted:6 August 2012
Published:10 August 2012

© 2012 Gao et al.; 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

Working in a weighted Sobolev space, a new result involving superlinear nonlinearities for a quasilinear elliptic boundary value problem in a domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M1">View MathML</a> is established. The proofs rely on the Galerkin method, Brouwer’s theorem and a new weighted compact Sobolev-type embedding theorem due to V.L. Shapiro.

MSC: 35J25, 35J62, 65L60.

Keywords:
weighted Sobolev space; superlinear; quasilinear elliptic equation

1 Introduction

Consider the following quasilinear elliptic problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M2">View MathML</a>

(1.1)

where Ω is an open (possibly unbounded) set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M1">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M4">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M5">View MathML</a> is the first eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M6">View MathML</a> ((2.3) below), and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M7">View MathML</a> is a singular quasilinear elliptic operator defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M8">View MathML</a>

(1.2)

The nonlinear part <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M9">View MathML</a> in Eq. (1.1) satisfies certain superlinear conditions.

There have been many results for quasilinear elliptic equations under the conditions of which the nonlinearities satisfy sublinear or linear growth in a weighted Sobolev space. One can refer to [1-6].

However, there seem to be relatively few papers that consider the quasilinear elliptic equations with superlinearity, because the compactly embedding theorem cannot be obtained easily.

The aim of this paper is to obtain an existence result for problem (1.1). Our methods combine the Galerkin-type techniques, Brouwer’s fixed-point theorem, and a new compactly embedding theorem established by V.L. Shapiro in [7].

This paper is organized as follows. In Section 2, we introduce some necessary assumptions and main results. In Section 3, four fundamental lemmas are established. In Section 4, the proofs of the main results are given.

2 Assumptions and main results

In this section, we introduce some assumptions and give the main results in this paper.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M10">View MathML</a> be a fixed closed set (it may be the empty set) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M11">View MathML</a> be weight functions. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M12">View MathML</a> is nonnegative (maybe identically zero). Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M13">View MathML</a> as the vector function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M14">View MathML</a>.

Consider the following pre-Hilbert spaces

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M15">View MathML</a>

with inner product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M17">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M18">View MathML</a>

with the inner product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M19">View MathML</a>

(2.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M20">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M22">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M23">View MathML</a> be the Hilbert space obtained through the completion of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M24">View MathML</a> by using the method of Cauchy sequences with respect to the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M25">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M26">View MathML</a> be the completion of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M27">View MathML</a> with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M28">View MathML</a>. Similarly, we may have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M29">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M22">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M31">View MathML</a>. Consequently, (2.1) may lead to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M32">View MathML</a>

(2.2)

Definition 2.1 For the quasilinear differential operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M7">View MathML</a>, the two-form is

For the linear differential operator,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M35">View MathML</a>

(2.3)

the two-form is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M36">View MathML</a>

Definition 2.2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M37">View MathML</a> is a simple-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M38">View MathML</a> region if the following conditions (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M39">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M40">View MathML</a>) hold:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M39">View MathML</a>) There exists a complete orthonormal system <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M42">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M43">View MathML</a>. Also, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M44">View MathML</a>, ∀n;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M45">View MathML</a>) There exists a sequence of eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M46">View MathML</a>, corresponding to the orthonormal sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M42">View MathML</a>, and satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M48">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M49">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M51">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M52">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M53">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M54">View MathML</a> is an open set for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M22">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M56">View MathML</a>) For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M57">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M58">View MathML</a> in (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M39">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M45">View MathML</a>), associated with each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M61">View MathML</a> there are positive functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M62">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M63">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M65">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M22">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M40">View MathML</a>) For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M61">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M69">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M70">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M71">View MathML</a>), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M72">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M73">View MathML</a> with the property

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M74">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M75">View MathML</a>.

There are many examples to illustrate the Simple-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M38">View MathML</a> region. One can refer to [7] and [8].

Remark 2.1 From (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M52">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M56">View MathML</a>), it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M57">View MathML</a> are positive and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M81">View MathML</a>

(2.4)

Definition 2.3<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M7">View MathML</a> is near-related to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M6">View MathML</a> if the following condition holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M84">View MathML</a>

We make the following assumptions concerning the operators <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M85">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M6">View MathML</a>: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M87">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M88">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M89">View MathML</a> satisfy (so do <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M90">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M91">View MathML</a>):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M92">View MathML</a>

(2.5)

It is assumed throughout the paper that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M93">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M94">View MathML</a>) meets:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M95">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M96">View MathML</a> is weakly sequentially continuous;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M97">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M98">View MathML</a>, s.t. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M100">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M101">View MathML</a> meets the following conditions:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M102">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M101">View MathML</a> satisfies the Caratheodory conditions;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M104">View MathML</a>) (superlinear growth condition) There exists θ with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M105">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M106">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M107">View MathML</a>. K is a nonnegative constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M108">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M109">View MathML</a>) There exists a nonnegative function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M110">View MathML</a> and a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M111">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M112">View MathML</a>

Remark 2.2 Observing that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M114">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M115">View MathML</a> is a positive function, and meets both (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M104">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M109">View MathML</a>).

Now we state our main results in this paper.

Theorem 2.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M37">View MathML</a>is a simple-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M38">View MathML</a>region, the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M7">View MathML</a>satisfies (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M95">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M97">View MathML</a>), and is near-related to the operator<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M6">View MathML</a>, (2.5) holds for both<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M7">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M6">View MathML</a>, fmeets (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M102">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M109">View MathML</a>), and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M128">View MathML</a> (the dual of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M129">View MathML</a>). Then the problem (1.1) has at least one nontrivial weak solution, that is, there exists a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M130">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M131">View MathML</a>

To derive out Theorem 2.1, we first discuss the problem in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M132">View MathML</a>, which is the subspace of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M26">View MathML</a> spanned by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M134">View MathML</a>. Then by virtue of the Galerkin method, the results will be extended to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M135">View MathML</a>.

3 Fundamental lemmas

In this section, we introduce and establish four fundamental lemmas. Lemmas 3.1 and 3.2 give two useful embedding theorems. Lemma 3.3 constructs some approximation solutions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M132">View MathML</a>. Lemma 3.4 studies the properties of the approximation solutions.

Lemma 3.1 ([7])

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M6">View MathML</a>is given by (2.3) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M138">View MathML</a>is a simple-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M38">View MathML</a>region. For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M140">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M141">View MathML</a>is compactly imbedded in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M142">View MathML</a>forθ (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M105">View MathML</a>); for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M144">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M26">View MathML</a>is compactly imbedded in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M146">View MathML</a>forθ (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M73">View MathML</a>).

Lemma 3.2 ([7])

Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M6">View MathML</a>is given by (2.3) and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M138">View MathML</a>is a simple-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M38">View MathML</a>region. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M26">View MathML</a>is compactly imbedded in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M23">View MathML</a>.

Lemma 3.3Let all the assumptions in Theorem 2.1 hold. Then for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M153">View MathML</a>, there exists a<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M154">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M155">View MathML</a>

(3.1)

Proof For fixed n (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M153">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M157">View MathML</a>, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M158">View MathML</a>. From simple-<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M38">View MathML</a> conditions, we obtain

(3.2)

(3.3)

From (3) and (4) of (2.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M162">View MathML</a>

(3.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M163">View MathML</a>. Combining (3.3) with (3.4), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M164">View MathML</a>

(3.5)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M165">View MathML</a>, a positive integer, we put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M166">View MathML</a>

(3.6)

Note from (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M104">View MathML</a>) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M168">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M169">View MathML</a>, a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M170">View MathML</a>. Also, from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M171">View MathML</a>, the Hölder inequality, Minkowski inequality, and (2.4), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M172">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M173">View MathML</a>

(3.7)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M174">View MathML</a> is a positive constant depending on m.

The remaining proof is separated into two parts. The first part is to prove the claim (3.8) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M175">View MathML</a>. The second part is to get the conclusion by leaving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M176">View MathML</a> based on (3.8).

Part 1. Fix m (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M165">View MathML</a>). To show there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M178">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M179">View MathML</a>

(3.8)

we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M180">View MathML</a>

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M181">View MathML</a>, where

(3.9)

(3.10)

For (3.9), observing the fact that the operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M7">View MathML</a> is near-related to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M185">View MathML</a>, (3.2), (3.5), (3.7), and Lemma 3.1, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M186">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M187">View MathML</a>

For (3.10), by (3.2), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M188">View MathML</a>

(3.11)

Consequently, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M189">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M190">View MathML</a> (here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M191">View MathML</a> is a large enough constant). By virtue of the generalized Brouwer’s theorem [9], there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M192">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M193">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M194">View MathML</a>. Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M195">View MathML</a>, then (3.8) holds.

Part 2. We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M196">View MathML</a> (n fixed) is uniformly bounded according to m.

Arguing by contradiction, without loss of generality, suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M197">View MathML</a>

(3.12)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M198">View MathML</a> in (3.8),

(3.13)

holds, that is,

(3.14)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M201">View MathML</a>.

On the other hand, using (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M109">View MathML</a>), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M203">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M204">View MathML</a>

(3.15)

Similarly, we can also obtain the same conclusion where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M205">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M206">View MathML</a>. As a result,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M207">View MathML</a>

(3.16)

(3.14) and (3.16) imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M208">View MathML</a>

(3.17)

Dividing both sides of (3.17) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M209">View MathML</a> and leaving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M176">View MathML</a>, we obtain from the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M7">View MathML</a> is near-related to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M212">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M213">View MathML</a>, and (3.5) together with Lemma 3.1 that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M214">View MathML</a>. However, n is a positive integer. So, we have arrived at a contradiction. (3.12) does not hold. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M215">View MathML</a>

(3.18)

(3.5) and (3.18) imply that there is a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M216">View MathML</a> (for ease of notation take the full sequence) and a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M217">View MathML</a>[10] such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M218">View MathML</a>

(3.19)

Therefore, from (3.19), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M219">View MathML</a>

(3.20)

And recall Lemma 3.1 that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M220">View MathML</a>

(3.21)

Moreover, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M221">View MathML</a> and a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M222">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M223">View MathML</a>, a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M170">View MathML</a> for ∀j.

Since by virtue of the Hölder inequality, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M102">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M104">View MathML</a>) and the Lebesgue dominated convergence theorem, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M227">View MathML</a>

Now replacing m by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M228">View MathML</a> in (3.8) and taking the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M229">View MathML</a> on both sides of the equation, we consequently obtain that (3.1) holds and Lemma 3.3 is completed. □

Lemma 3.4Let all the assumptions in Theorem 2.1 hold. Then the sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M230">View MathML</a>obtained in Lemma 3.3 is uniformly bounded in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M231">View MathML</a>.

Proof For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M232">View MathML</a> in Lemma 3.3, set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M233">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M234">View MathML</a>.

We suppose that Lemma 3.4 is false. Without loss of generality, suppose that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M235">View MathML</a>

(3.22)

To lead to a contradiction, taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M236">View MathML</a> in (3.1), then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M237">View MathML</a>

(3.23)

And we can get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M238">View MathML</a>

(3.24)

In view of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M239">View MathML</a>, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M109">View MathML</a>), and (3.24),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M241">View MathML</a>

(3.25)

Dividing both sides of (3.25) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M242">View MathML</a> and leaving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M49">View MathML</a>, from the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M7">View MathML</a> is near-related to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M245">View MathML</a> and Lemma 3.1, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M246">View MathML</a>

(3.26)

Apply (3) and (4) of (2.5) in conjunction with (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M97">View MathML</a>) to show

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M248">View MathML</a>

(3.27)

also, (1) and (2) of (2.5) to show

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M249">View MathML</a>

(3.28)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M250">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M251">View MathML</a> are positive constants. Setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M252">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M253">View MathML</a>, it is obvious <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M254">View MathML</a>. By (3.27) and (3.28), we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M255">View MathML</a>

(3.29)

Using (3.29) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M109">View MathML</a>), from (3.25), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M257">View MathML</a>

(3.30)

From (3.4), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M258">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M259">View MathML</a>. It is easy to obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M260">View MathML</a>

(3.31)

Dividing both sides of (3.31) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M261">View MathML</a> and leaving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M49">View MathML</a>, by (3.26), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M263">View MathML</a>

(3.32)

So, we have arrived at a contradiction. Thus, there holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M264">View MathML</a>

(3.33)

Lemma 3.4 is completed. □

4 Proof of Theorem 2.1

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M26">View MathML</a> is a separable Hilbert space, from Lemma 3.1 and Lemma 3.2, we conclude that there exist a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M266">View MathML</a> (which for ease of notation we take the full sequence) and a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M267">View MathML</a> with the following properties [10]:

(4.1)

(4.2)

(4.3)

(4.4)

(4.5)

(4.6)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M274">View MathML</a> where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M275">View MathML</a> is a fixed but arbitrary positive integer. In fact, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M276">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M277">View MathML</a>

(4.7)

Observing (1) of (2.5), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M95">View MathML</a>)-(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M97">View MathML</a>), (4.3)-(4.5) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M280">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M281">View MathML</a>

(4.8)

On the other hand, applying (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M102">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M104">View MathML</a>), (4.2), (4.3), the Hölder inequality, and the Lebesgue dominated convergence theorem, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M284">View MathML</a>

(4.9)

Also, by (3.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M285">View MathML</a>

(4.10)

For (4.10), leaving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M49">View MathML</a>, from (4.1), (4.8), and (4.9), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M287">View MathML</a>

(4.11)

Next, given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M288">View MathML</a>, we define a projection <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M289">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M290">View MathML</a>

(4.12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M291">View MathML</a>. It is easy to get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M292">View MathML</a>. As a result, there hold

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M293">View MathML</a>

(4.13)

Replacing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M294">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M295">View MathML</a> in (4.11), passing to the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M296">View MathML</a> on both sides, and using (4.13), we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/90/mathml/M297">View MathML</a>

Hence, the proof of Theorem 2.1 is complete. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The paper is the result of joint work of all authors who contributed equally to the final version of the paper. All authors read and approved the final manuscript.

Acknowledgements

The authors express their sincere thanks to the referees for their valuable suggestions. This work was supported financially by the National Natural Science Foundation of China (11171220).

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