Open Access Research

Existence of positive solutions of elliptic mixed boundary value problem

Guofa Li

Author Affiliations

Department of Mathematics and Information Science, Qujing Normal University, Qujing, 655011, P.R. China

Boundary Value Problems 2012, 2012:91  doi:10.1186/1687-2770-2012-91


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/91


Received:19 January 2012
Accepted:6 August 2012
Published:16 August 2012

© 2012 Li; 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 use variational methods to prove two existence of positive solutions of the following mixed boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M1">View MathML</a>

One deals with the asymptotic behaviors of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M2">View MathML</a> near zero and infinity and the other deals with superlinear of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M2">View MathML</a> at infinity.

MSC: 35M12, 35D30.

Keywords:
elliptic mixed boundary value problem; positive solutions; mountain pass theorem; Sobolev embedding theorem

1 Introduction and preliminaries

This paper is concerned with the existence of positive solutions of the following elliptic mixed boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M4">View MathML</a>

(1)

where Ω is a bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M5">View MathML</a> with Lipschitz boundary Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M7">View MathML</a>, Γ is a sufficiently smooth <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M8">View MathML</a>-dimensional manifold, and ν is the outward normal vector on Ω. We assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M10">View MathML</a> are continuous and satisfy

(S1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M14">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13">View MathML</a>.

(S2) For almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M19">View MathML</a> is nondecreasing with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M20">View MathML</a>.

(S3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M22">View MathML</a> uniformly in a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M25">View MathML</a> is the first eigenvalue of (2), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M27">View MathML</a>.

(S4) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M28">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M29">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M30">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M32">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M33">View MathML</a>.

The eigenvalue problem of (1) is studied by Liu and Su in [1]

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M34">View MathML</a>

(2)

There exists a set of eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M35">View MathML</a> and corresponding eigenfunctions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M36">View MathML</a> which solve problem (2), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M38">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M40">View MathML</a>.

There have been many papers concerned with similar problems at resonance under the boundary condition; see [2-10]. Moreover, some multiplicity theorems are obtained by the topological degree technique and variational methods; interested readers can see [11-17]. Problem (1) is different from the classical ones, such as those with Dirichlet, Neuman, Robin, No-flux, or Steklov boundary conditions.

In this paper, we assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M41">View MathML</a> is a closed subspace of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M42">View MathML</a>. We define the norm in V as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M44">View MathML</a> is the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M45">View MathML</a> norm, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M46">View MathML</a> is the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M47">View MathML</a> norm, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M48">View MathML</a> is the trace operator with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M49">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M50">View MathML</a>, that is continuous and compact (see [18]). Furthermore, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M52">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M20">View MathML</a> (see [1]). Then, by (S3), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M54">View MathML</a>

(3)

Let Ω be a bounded domain with a Lipschitz boundary; there is a continuous embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M55">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M56">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M31">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M58">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M59">View MathML</a>. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M60">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M61">View MathML</a>

(4)

Moreover, there is a continuous boundary trace embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M62">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M63">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M31">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M65">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M33">View MathML</a>. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M67">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M68">View MathML</a>

(5)

It is well known that to seek a nontrivial weak solution of problem (1) is equivalent to finding a nonzero critical value of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M69">View MathML</a> functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M70">View MathML</a>

(6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M73">View MathML</a>. Moreover, by (S1) and the Strong maximum principle, a nonzero critical point of J is in fact a positive solution of (1). In order to find critical points of the functional (6), one often requires the technique condition, that is, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M75">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M77">View MathML</a>

(AR)

It is easy to see that the condition (AR) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M78">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M79">View MathML</a> must be superlinear with respect to u at infinity. In the present paper, motivated by [19] and [20], we study the existence and nonexistence of positive solutions for problem (1) with the asymptotic behavior assumptions (S3) of f at zero and infinity. Moreover, we also study superlinear of f at infinity with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M80">View MathML</a> in (S3), which is weaker than the (AR) condition, that is the (AR) condition does not hold.

In order to get our conclusion, we define the minimization problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M81">View MathML</a>

(7)

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M82">View MathML</a>, which is achieved by some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M83">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M84">View MathML</a> a.e. in Ω; see Lemma 1.

We denote by c, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M86">View MathML</a> universal constants unless specified otherwise. Our main results are as follows.

Theorem 1Let conditions (S1) to (S3) hold, then:

(i) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M87">View MathML</a>, then the problem (1) has no any positive solution inV.

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M88">View MathML</a>, then the problem (1) has at least one positive solution inV.

(iii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M89">View MathML</a>, then the problem (1) has one positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M90">View MathML</a>if and only if there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M91">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M92">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M94">View MathML</a>a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M84">View MathML</a>is the function which achieves Λ.

Corollary 2Let conditions (S1) to (S3) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M97">View MathML</a>hold, then:

(i) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M98">View MathML</a>, then the problem (1) has no any positive solution inV.

(ii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M99">View MathML</a>, then the problem (1) has at least one positive solution inV.

(iii) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M100">View MathML</a>, then the problem (1) has one positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M90">View MathML</a>if and only if there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M91">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M103">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M105">View MathML</a>a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M107">View MathML</a>is the eigenfunction of the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M25">View MathML</a>.

Theorem 3Let conditions (S1) to (S4) with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M80">View MathML</a>hold, then the problem (1) has at least one positive solution inV.

2 Some lemmas

We need the following lemmas.

Lemma 1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M111">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M112">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M82">View MathML</a>and there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M114">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M115">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M116">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M84">View MathML</a>a.e. inV.

Proof By the Sobolev embedding function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M118">View MathML</a> and Fatou’s lemma, it is easy to know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M82">View MathML</a> and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M114">View MathML</a>, which satisfies Λ, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M121">View MathML</a>. Furthermore, we assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M122">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M123">View MathML</a> could replace by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M124">View MathML</a>. By the Strong maximum principle, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M84">View MathML</a> a.e. in V. □

Lemma 2If conditions (S1) to (S3) hold, then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M126">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M129">View MathML</a>.

Proof By condition (S3), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M131">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M132">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M133">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M134">View MathML</a>. Which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M136">View MathML</a>.

By (4) and (5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M137">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M138">View MathML</a>; we take ε which satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M139">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M140">View MathML</a>. Then we take a positive constant β such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M127">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M129">View MathML</a>, and is small enough. □

Lemma 3If conditions (S1) to (S3) hold, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M84">View MathML</a>is defined by Lemma 1, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M145">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M146">View MathML</a>.

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M84">View MathML</a> is defined by Lemma 1, by Fatou’s lemma, and (S3), we have

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M145">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M146">View MathML</a>. □

Lemma 4Let conditions (S1) and (S2) hold. If a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M152">View MathML</a>satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M153">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M154">View MathML</a>, then there exists a subsequence of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M155">View MathML</a>, still denoted by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M155">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M157">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M159">View MathML</a>.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M160">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M154">View MathML</a>, for a subsequence, we may assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M162">View MathML</a>

(8)

For any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M159">View MathML</a>, set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M165">View MathML</a>

Then (S2) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M166">View MathML</a>

It implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M167">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M168">View MathML</a>. Following the same procedures, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M169">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M168">View MathML</a>.

For all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M20">View MathML</a> and positive integer n, by (8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M172">View MathML</a>

(9)

On the other hand, by (8), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M173">View MathML</a>

One has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M174">View MathML</a>

(10)

Combining (9) and (10), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M175">View MathML</a>. □

Lemma 5 (see [21])

SupposeEis a real Banach space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M176">View MathML</a>satisfies the following geometrical conditions:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M177">View MathML</a>; there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M178">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M179">View MathML</a>;

(ii) There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M180">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M181">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M182">View MathML</a>be the set of all continuous paths joining 0 ande:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M183">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M184">View MathML</a>

Then there exists a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M185">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M186">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M187">View MathML</a>.

3 Proofs of main results

Proof of Theorem 1 (i) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M71">View MathML</a> is one positive solution of problem (1), by (3), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M189">View MathML</a>

That is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M190">View MathML</a>

It implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M191">View MathML</a>. This completes the proof of Theorem 1(i).

(ii) By Lemma 2, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M192">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M127">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M129">View MathML</a>. By Lemma 3, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M195">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M196">View MathML</a>. Define

(11)

(12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M84">View MathML</a> is given by Lemma 1. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M200">View MathML</a> and by Lemma 3, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M152">View MathML</a> such that

(13)

(14)

(14) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M204">View MathML</a>

(15)

Here, in what follows, we use <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M205">View MathML</a> to denote any quantity which tends to zero as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M154">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M155">View MathML</a> is bounded in V, when Ω is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M209">View MathML</a> are subcritical, we can get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M155">View MathML</a> has a subsequence strong convergence to a critical value of J, and our proof is complete. So, to prove the theorem, we only need show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M155">View MathML</a> is bounded in V. Supposing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M212">View MathML</a> is unbounded, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M213">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M154">View MathML</a>. We order

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M215">View MathML</a>

(16)

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M216">View MathML</a> is bounded in V. By extracting a subsequence, we suppose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M217">View MathML</a> is a strong convergence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M217">View MathML</a> is a convergence a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M221">View MathML</a> is a weak convergence in V.

We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M222">View MathML</a>. In fact, by (S1) and (S3), we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M223">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M224">View MathML</a>, and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M225">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M226">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M227">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M228">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M229">View MathML</a> is a strong convergence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M218">View MathML</a>, and by (15) and (16) we know

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M231">View MathML</a>

It is contradiction with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M91">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M222">View MathML</a>.

As follows, we prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M222">View MathML</a> satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M235">View MathML</a>

We order

By (S1) and (S3), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M237">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M238">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M239">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M240">View MathML</a>. We select a suitable subsequence and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M241">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M242">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M243">View MathML</a> is a strong convergence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M245">View MathML</a> is a strong convergence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M246">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M248">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M240">View MathML</a>.

It follows from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M217">View MathML</a> is a strong convergence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M218">View MathML</a> that

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M253">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M255">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M218">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M257">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M246">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M259">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M246">View MathML</a>.

By (16), we have

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M221">View MathML</a> is a weak convergence in V, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M263">View MathML</a>

We order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M264">View MathML</a>; this yields <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M265">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M266">View MathML</a>. By the Strong maximum principle, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M267">View MathML</a> a.e. in Ω, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M268">View MathML</a> a.e. in Ω. Combining (S3) and (3), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M269">View MathML</a>

This is a contradiction with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M88">View MathML</a>. This completes the proof of Theorem 1(ii).

(iii) If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M89">View MathML</a>, by Lemma 1, there exists some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M84">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M273">View MathML</a>

(17)

If u is a positive solution of (1), for the above <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M123">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M275">View MathML</a>

(18)

We order <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M276">View MathML</a> in (17), and it follows from (18) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M277">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M278">View MathML</a>.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M84">View MathML</a> a.e. in Ω, combining (S2), (S3), and (3), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M280">View MathML</a>

Then we must have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M282">View MathML</a> a.e. in Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M283">View MathML</a> also achieves Λ (=1). When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M284">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M91">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M286">View MathML</a>, which achieves Λ.

On the other hand, if for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M91">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M92">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M289">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M290">View MathML</a> a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13">View MathML</a>, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M292">View MathML</a> also achieves Λ. This means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M92">View MathML</a> is a solution of problem (1) as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M89">View MathML</a>. This completes the proof of Theorem 1(iii). □

Proof of Corollary 2 Note that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M295">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M296">View MathML</a>. The conclusion follows from Theorem 1. □

Proof of Theorem 3 When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M80">View MathML</a>, we can replace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M298">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M299">View MathML</a> in (11) and define c as in (12), then following the same procedures as in the proof of Theorem 1(ii), we need to show only that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M155">View MathML</a> is bounded in V. For this purpose, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M216">View MathML</a> be defined as in (16). If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M216">View MathML</a> is bounded in V, we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M217">View MathML</a> is a strong convergence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M217">View MathML</a> is convergence a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M221">View MathML</a> is a weak convergence in V, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M308">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M213">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M310">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M311">View MathML</a>. We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M312">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M313">View MathML</a>. Obviously, by (16), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M314">View MathML</a> a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M315">View MathML</a>. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M80">View MathML</a> in (S3), there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M317">View MathML</a> and n large enough we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M318">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M319">View MathML</a> uniformly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M320">View MathML</a>. Hence, by (15) and (16), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M321">View MathML</a>

Noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M322">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M315">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M324">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M325">View MathML</a> can be chosen large enough, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M326">View MathML</a> and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M311">View MathML</a> in Ω.

Then we know <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M328">View MathML</a>, and consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M329">View MathML</a>

(19)

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M213">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M310">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M154">View MathML</a>, then it follows Lemma 4 and (13), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M333">View MathML</a>

(20)

Obviously, (19) and (20) are contradictory. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M155">View MathML</a> is bounded in V. This completes the proof of Theorem 3. □

4 Example

In this section, we give two examples on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M79">View MathML</a>: One satisfies (S1) to (S3) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M80">View MathML</a>, but does not satisfy the (AR) condition; the other illustrates how the assumptions on the boundary are not trivial and compatible with the inner assumptions in Ω.

Example 1 Set:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M337">View MathML</a>

Then it is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M338">View MathML</a> satisfies (S1) to (S3) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M339">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M340">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M341">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M146">View MathML</a>. In addition,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M343">View MathML</a>

So, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M345">View MathML</a>, for all t large.

This means <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M338">View MathML</a> does not satisfy the (AR) condition.

Example 2 Consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M347">View MathML</a>

(21)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M348">View MathML</a> is a constant. It is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M51">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M350">View MathML</a>. Problem (21) is a case of (1); we can obtain the nontrivial solution: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M351">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/91/mathml/M352">View MathML</a>.

Competing interests

The author declares that he has no competing interests.

Author’s contributions

Li G carried out all studies in this article.

Acknowledgements

The author would like to thank the referees for carefully reading this article and making valuable comments and suggestions.

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