Open Access Research

On semilinear biharmonic equations with concave-convex nonlinearities involving weight functions

Lu Yang12* and Xuan Wang3

Author Affiliations

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

2 Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu, 730000, P.R. China

3 College of Mathematics and Statistics, Northwest Normal University, Lanzhou, 730070, P.R. China

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Boundary Value Problems 2014, 2014:117  doi:10.1186/1687-2770-2014-117


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/117


Received:1 March 2014
Accepted:29 April 2014
Published:14 May 2014

© 2014 Yang and Wang; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

In this paper, we consider semilinear biharmonic equations with concave-convex nonlinearities involving weight functions, where the concave nonlinear term is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M1">View MathML</a> and the convex nonlinear term is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M2">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M3">View MathML</a>. By use of the Nehari manifold and the direct variational methods, the existence of multiple positive solutions is established as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4">View MathML</a>, here the explicit expression of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M5">View MathML</a> is provided.

MSC: 35J35, 35J40, 35J65.

Keywords:
biharmonic equations; concave-convex nonlinearities; weight functions

1 Introduction

In recent years, there has been extensive attention on semilinear second-order elliptic equations,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M6">View MathML</a>

(1.1)

here Ω is a bounded smooth domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M7">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M8">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M9">View MathML</a> and λ is a positive parameter; see [1-8] and the references therein. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M10">View MathML</a> is sublinear, say, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M12">View MathML</a>, the monotone iteration scheme or the method of sub-solutions and super-solutions are effective; see [9]. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M10">View MathML</a> is superlinear, for example, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M15">View MathML</a>, variational methods are applicable; see [10]. In contrast with the pure sublinear case and the pure superlinear case, in [2] Ambrosetti et al. considered problem (1.1) when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M10">View MathML</a> is, roughly, the sum of a sublinear and a superlinear term. To be precise, they considered the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M17">View MathML</a>

(1.2)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M18">View MathML</a>. They proved that problem (1.2) admits at least two positive solutions for λ sufficiently small. In [6], Sun and Li considered a similar problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M19">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M20">View MathML</a>, the authors studied the value of Λ, the supremum of the set λ, related to the existence and multiplicity of positive solutions and established uniform lower bounds for Λ. In [8], Wu considered the subcritical case of problem (1.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M21">View MathML</a> replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M22">View MathML</a>, here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M23">View MathML</a> is a sign-changing function, and he showed that problem (1.2) has at least two positive solutions as λ is small enough.

Some interesting generalizations of (1.2) have been provided in the framework of quasi-linear elliptic equations or systems, semilinear second-order elliptic systems or fourth-order elliptic equations. More recently, the semilinear fourth-order elliptic equations have been studied by many authors, we refer the reader to [11-13] and the references therein. Motivated by some work in [6,8,13], we deal with the following semilinear biharmonic elliptic equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M24">View MathML</a>

(1.3)

where Ω is a bounded smooth domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M7">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M26">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M27">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M28">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M30">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M31">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M32">View MathML</a> is a parameter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M33">View MathML</a> is a positive or sign-changing weight function and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M34">View MathML</a> is a positive weight function.

For convenience and simplicity, we introduce some notations. The norm of u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M35">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M36">View MathML</a>, the norm of u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M37">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M38">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M39">View MathML</a> is denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M40">View MathML</a>, endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M41">View MathML</a>; S denotes the best Sobolev constant for the embedding of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M40">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M43">View MathML</a> (see [14]); to be precise, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M44">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M45">View MathML</a>.

Now we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M46">View MathML</a>

It is well known that the weak solutions of problem (1.3) are the critical points of the energy functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47">View MathML</a> (see Rabinowitz [15]).

Next, we consider the Nehari minimization problem: for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M48">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M49">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M50">View MathML</a>. Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M51">View MathML</a>

Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M52">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M53">View MathML</a>

Similarly to the method used in Tarantello [16], we split <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M54">View MathML</a> into three parts:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M55">View MathML</a>

Note that all solutions of (1.3) are clearly in the Nehari manifold, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M54">View MathML</a>. Hence, our approach to solve problem (1.3) is to analyze the structure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M54">View MathML</a>, and then to deal with the minimization problems for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M59">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M60">View MathML</a> applying the direct variational method.

The following is our main result.

Theorem 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M61">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M62">View MathML</a>, then problem (1.3) has at least two positive solutions for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4">View MathML</a>.

The paper is organized as follows: in Section 2, we give some lemmas; in Section 3, we prove Theorem 1.1.

2 Preliminaries

In this section, we prove several lemmas.

Lemma 2.1For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4">View MathML</a> (where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M65">View MathML</a>is given in Theorem 1.1), we have<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M66">View MathML</a>.

Proof Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M67">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M32">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M69">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M70">View MathML</a>

(2.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M71">View MathML</a>

(2.2)

By (2.1)-(2.2), the Sobolev inequality, and the Hölder inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M72">View MathML</a>

(2.3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M73">View MathML</a>

(2.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M74">View MathML</a>. Thus, using (2.3) and (2.4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M75">View MathML</a>

(2.5)

Hence, by (2.5) the desired conclusion yields. □

Lemma 2.2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M76">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M77">View MathML</a>

Proof From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M76">View MathML</a>, it is easy to see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M79">View MathML</a>

By the Sobolev inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M80">View MathML</a>

In addition,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M81">View MathML</a>

The proof is completed. □

By Lemma 2.1, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4">View MathML</a> we write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M83">View MathML</a> and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M84">View MathML</a>

The following lemma shows that the minimizers on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M54">View MathML</a> are ‘usually’ critical points for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47">View MathML</a>.

Lemma 2.3For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4">View MathML</a>, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M88">View MathML</a>is a local minimizer for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47">View MathML</a>on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M54">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M91">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M92">View MathML</a>.

Proof If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M88">View MathML</a> is a local minimizer for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M54">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M88">View MathML</a> is a solution of the optimization problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M97">View MathML</a>

Hence, by the theory of Lagrange multipliers, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M98">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M99">View MathML</a>

(2.6)

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M100">View MathML</a>

(2.7)

From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M101">View MathML</a> and Lemma 2.1, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M103">View MathML</a>. So, by (2.6)-(2.7) we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M104">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M105">View MathML</a>. □

For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M106">View MathML</a>, we write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M107">View MathML</a>

Then we have the following lemma.

Lemma 2.4For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M106">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M109">View MathML</a>, we have

(i) there is a unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M110">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M111">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M112">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M113">View MathML</a>is a continuous function for nonzerou;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M114">View MathML</a>;

(iv) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M115">View MathML</a>, then there is a unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M116">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M117">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M118">View MathML</a>.

Proof (i) Fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M119">View MathML</a>. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M120">View MathML</a>

Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M122">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M124">View MathML</a> is concave and reaches its maximum at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M125">View MathML</a>. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M126">View MathML</a>

(2.8)

Case I. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M127">View MathML</a>.

There is a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M128">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M129">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M130">View MathML</a>. Now,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M131">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M132">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M133">View MathML</a>. In addition,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M134">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M135">View MathML</a>

Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M136">View MathML</a>.

Case II. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M115">View MathML</a>.

From (2.8) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M138">View MathML</a>

there exist unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M139">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M140">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M141">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M142">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M143">View MathML</a>

Similar to the argument in Case I above, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M144">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M133">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M146">View MathML</a>

(ii) By the uniqueness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M113">View MathML</a> and the external property of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M113">View MathML</a>, we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M113">View MathML</a> is continuous function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M150">View MathML</a>.

(iii) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M151">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M152">View MathML</a>. By item (i), there is a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M153">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M154">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M155">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M151">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M157">View MathML</a>, which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M158">View MathML</a>

Conversely, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M159">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M160">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M161">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M162">View MathML</a>

(iv) By Case II of item (i). □

By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M33">View MathML</a> and changes sign in Ω, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M164">View MathML</a> is an open set in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M165">View MathML</a>. Without loss of generality, we may assume that Θ is a domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M165">View MathML</a>. Consider the following biharmonic equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M167">View MathML</a>

(2.9)

Associated with (2.9), we consider the energy functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M168">View MathML</a>

and the minimization problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M169">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M170">View MathML</a>. Now we prove that problem (2.9) has a positive solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M171">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M172">View MathML</a>.

Lemma 2.5For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M173">View MathML</a>, there exists a unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M174">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M175">View MathML</a>. The maximum of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M176">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M177">View MathML</a>is reached at<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M178">View MathML</a>, the map

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M179">View MathML</a>

is continuous and the induced continuous map<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M180">View MathML</a>defines a homeomorphism of the unit sphere of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M181">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M182">View MathML</a>.

Proof For any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M183">View MathML</a>, consider the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M184">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M177">View MathML</a>. Clearly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M186">View MathML</a>

(2.10)

It is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M187">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M188">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M189">View MathML</a> small and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M190">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M189">View MathML</a> large. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M192">View MathML</a> is reached at a unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M178">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M194">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M195">View MathML</a>. To prove the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M196">View MathML</a>, assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M197">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M198">View MathML</a>. It is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M199">View MathML</a> is bounded. If a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M199">View MathML</a> converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M201">View MathML</a>, it follows from (2.10) that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M202">View MathML</a> and then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M203">View MathML</a>. Finally the continuous map from the unit sphere of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M181">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M182">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M180">View MathML</a>, is inverse to the retraction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M207">View MathML</a>. □

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M208">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M209">View MathML</a>.

Lemma 2.6<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M210">View MathML</a>is a critical value ofK.

Proof From Lemma 2.5, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M211">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M212">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M213">View MathML</a> and t large, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M214">View MathML</a>. The manifold <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M182">View MathML</a> separates <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M181">View MathML</a> into two components. The component containing the origin also contains a small ball around the origin. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M217">View MathML</a> for all u in this component, because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M218">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M219">View MathML</a>. Then each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M220">View MathML</a> has to cross <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M182">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M222">View MathML</a>. Since the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M223">View MathML</a> is compact (see [14]), it is easy to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M224">View MathML</a> is a critical value of K and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M171">View MathML</a> a positive solution corresponding to c. □

With the help of Lemma 2.6, we have the following result.

Lemma 2.7 (i) For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4">View MathML</a>, there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M227">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M228">View MathML</a>

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47">View MathML</a>is coercive and bounded below on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M54">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M32">View MathML</a>.

Proof (i) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M171">View MathML</a> be a positive solution of problem (2.9) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M233">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M234">View MathML</a>

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M235">View MathML</a> as defined by Lemma 2.4(iv). Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M236">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M237">View MathML</a>

This implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M238">View MathML</a>

(ii) For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M52">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M240">View MathML</a>. Then by the Hölder, Sobolev, and Young inequalities,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M241">View MathML</a>

here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M242">View MathML</a>.

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47">View MathML</a> is coercive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M54">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M245">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M32">View MathML</a>. □

Next, we will use the idea of Tarantello [16] to get the following results.

Lemma 2.8For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4">View MathML</a>and any given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M248">View MathML</a>, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M249">View MathML</a>and a differentiable functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M250">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M251">View MathML</a>, the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M252">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M253">View MathML</a>

(2.11)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M254">View MathML</a>.

Proof Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M255">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M256">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M257">View MathML</a> and by Lemma 2.1, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M258">View MathML</a>

we can get the desired results applying the implicit function theorem at the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M259">View MathML</a>. □

Lemma 2.9For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4">View MathML</a>and any given<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M261">View MathML</a>, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M249">View MathML</a>and a differentiable functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M263">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M264">View MathML</a>, the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M265">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M266">View MathML</a>

(2.12)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M254">View MathML</a>.

Proof In view of Lemma 2.8, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M249">View MathML</a> and a differentiable functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M269">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M264">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M271">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M272">View MathML</a> and we have (2.12). By use of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M151">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M274">View MathML</a>. In combination with the continuity of the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M275">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M269">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M277">View MathML</a> as ϵ sufficiently small, this implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M278">View MathML</a>. □

3 Proof of Theorem 1.1

Firstly, we provide the existence of minimizing sequences for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M54">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M60">View MathML</a> as λ is sufficiently small.

Proposition 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4">View MathML</a>, then

(i) there exists a minimizing sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M283">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M284">View MathML</a>

(ii) there exists a minimizing sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M285">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M286">View MathML</a>

Proof (i) By Lemma 2.7(ii) and the Ekeland variational principle [17], there exists a minimizing sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M287">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M288">View MathML</a>

(3.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M289">View MathML</a>

(3.2)

Taking n large, from Lemma 2.7(i) and (3.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M290">View MathML</a>

(3.3)

This implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M291">View MathML</a>

(3.4)

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M292">View MathML</a>

(3.5)

Now, we will show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M293">View MathML</a>

Exactly as in Lemma 2.8 we may apply suitable functionals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M294">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M295">View MathML</a> and obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M296">View MathML</a>

(3.6)

Hence, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M297">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M298">View MathML</a> small, substituting in (3.6) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M299">View MathML</a> and applying (3.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M300">View MathML</a>

Dividing by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M298">View MathML</a> and passing to the limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M302">View MathML</a> we derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M303">View MathML</a>

(3.7)

Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M304">View MathML</a>

by the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M295">View MathML</a> we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M306">View MathML</a>

(3.8)

for a suitable positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M307">View MathML</a>.

Next, we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M308">View MathML</a> is bounded away from zero. Arguing by contradiction, assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M309">View MathML</a>

(3.9)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M310">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M311">View MathML</a>

and consequently by (3.9),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M312">View MathML</a>

(3.10)

Then by (3.4), the Hölder inequality, Sobolev inequality and (3.9)-(3.10), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M313">View MathML</a>

moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M314">View MathML</a>, which contradicts (3.5).

Thus, we get from (3.8) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M315">View MathML</a>

Hence, by (3.7) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M316">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M317">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M318">View MathML</a>.

(ii) Similar to the arguments in (i), by Lemma 2.9 and Lemma 2.2, we can prove (ii). □

Now, we establish the existence of a local minimum for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M320">View MathML</a>.

Theorem 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4">View MathML</a>, then the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47">View MathML</a>has a minimizer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M323">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M320">View MathML</a>and it satisfies

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M325">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M323">View MathML</a>is a positive solution of problem (1.3);

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M327">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M328">View MathML</a>.

Proof By Proposition 3.1(i), there is a minimizing sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M329">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M54">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M332">View MathML</a>

(3.11)

Then by Lemma 2.7 and the compact imbedding theorem, there exist a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M333">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M334">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M335">View MathML</a>

(3.12)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M336">View MathML</a>

(3.13)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M337">View MathML</a>

(3.14)

First, we claim that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M338">View MathML</a>

If not, by (3.14) we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M339">View MathML</a>

Therefore, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M340">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M341">View MathML</a>

this contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M342">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M340">View MathML</a>.

In combination with (3.11)-(3.14), it is easy to verify that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M344">View MathML</a> is a nontrivial weak solution of problem (1.3).

Now we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M345">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M346">View MathML</a>. Supposing the contrary, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M347">View MathML</a> and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M348">View MathML</a>

this contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M344">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M345">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M346">View MathML</a>. This implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M352">View MathML</a>

Moreover, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M353">View MathML</a>. In fact, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M354">View MathML</a>, by Lemma 2.4, there exist unique <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M355">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M356">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M357">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M358">View MathML</a>, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M359">View MathML</a>. Since

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M360">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M361">View MathML</a>

there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M362">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M363">View MathML</a>. By Lemma 2.4,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M364">View MathML</a>

which is a contradiction. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M365">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M366">View MathML</a>, by Lemma 2.3 we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M323">View MathML</a> is a nonnegative weak solution to problem (1.3). Applying the regularity theory and strong maximum principle of elliptic equations, we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M323">View MathML</a> is one positive solution of problem (1.3). In addition, by Lemma 2.7,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M369">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M327">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M328">View MathML</a>. □

Next, we establish the existence of a local minimum for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M60">View MathML</a>.

Theorem 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M4">View MathML</a>, then the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47">View MathML</a>has a minimizer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M376">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M60">View MathML</a>and it satisfies

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M378">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M376">View MathML</a>is a positive solution of problem (1.3).

Proof By Proposition 3.1(ii), there is a minimizing sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M329">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M47">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M60">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M383">View MathML</a>

Then by Lemma 2.7 and the compact imbedding theorem, there exist a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M333">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M385">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M386">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M387">View MathML</a>

Connecting with Lemma 2.2, it is easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M388">View MathML</a> is a nontrivial weak solution of problem (1.3).

Next we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M389">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M346">View MathML</a>. Supposing the contrary, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M391">View MathML</a> and so

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M392">View MathML</a>

this contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M388">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M389">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M346">View MathML</a>. This implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M396">View MathML</a>

In addition, from Lemma 2.4(ii)-(iii), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M397">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M398">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M399">View MathML</a>, by Lemma 2.3 we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M376">View MathML</a> is a nonnegative weak solution to problem (1.3). Applying the regularity theory and strong maximum principle of elliptic equations, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/117/mathml/M376">View MathML</a> is one positive solution of problem (1.3). □

Proof of Theorem 1.1 It is an immediate consequence of Theorems 3.1 and 3.2. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors read and approved the final manuscript.

Acknowledgements

This work is partly supported by NNSF (11101404, 11201204, 11361053) of China and the Young Teachers Scientific Research Ability Promotion Plan of Northwest Normal University (NWNU-LKQN-11-5).

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