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This article is part of the series Recent Advances in Operator Equations, Boundary Value Problems, Fixed Point Theory and Applications, and General Inequalities.

Open Access Research

Existence of a positive solution for quasilinear elliptic equations with nonlinearity including the gradient

Mieko Tanaka

Author Affiliations

Department of Mathematics, Tokyo University of Science, Kagurazaka 1-3, Shinjyuku-ku, Tokyo, 162-8601, Japan

Boundary Value Problems 2013, 2013:173  doi:10.1186/1687-2770-2013-173


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/173


Received:15 May 2013
Accepted:10 July 2013
Published:24 July 2013

© 2013 Tanaka; licensee Springer

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

Abstract

We provide the existence of a positive solution for the quasilinear elliptic equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M1">View MathML</a>

in Ω under the Dirichlet boundary condition. As a special case (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M2">View MathML</a>), our equation coincides with the usual p-Laplace equation. The solution is established as the limit of a sequence of positive solutions of approximate equations. The positivity of our solution follows from the behavior of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M3">View MathML</a> as t is small. In this paper, we do not impose the sign condition to the nonlinear term f.

MSC: 35J92, 35P30.

Keywords:
nonhomogeneous elliptic operator; positive solution; the first eigenvalue with weight; approximation

1 Introduction

In this paper, we consider the existence of a positive solution for the following quasilinear elliptic equation:

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M5">View MathML</a> is a bounded domain with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M6">View MathML</a> boundary Ω. Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M7">View MathML</a> is a map which is strictly monotone in the second variable and satisfies certain regularity conditions (see the following assumption (A)). Equation (P) contains the corresponding p-Laplacian problem as a special case. However, in general, we do not suppose that this operator is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M8">View MathML</a>-homogeneous in the second variable.

Throughout this paper, we assume that the mapAand the nonlinear termfsatisfy the following assumptions (A) and (f), respectively.

(A) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M9">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M10">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M11">View MathML</a>, and there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M17">View MathML</a> such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M18">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M19">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M20">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M21">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M22">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M21">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M25">View MathML</a>;

(iv) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M26">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M21">View MathML</a>;

(v) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M29">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M31">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M32">View MathML</a>.

(f) f is a continuous function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M33">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M34">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M35">View MathML</a> and the following growth condition: there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M37">View MathML</a> and a continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M38">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M39">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M40">View MathML</a>

(1)

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M41">View MathML</a>.

In this paper, we say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M42">View MathML</a> is a (weak) solution of (P) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M43">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M44">View MathML</a>.

A similar hypothesis to (A) is considered in the study of quasilinear elliptic problems (see [[1], Example 2.2.], [2-5] and also refer to [6,7] for the generalized p-Laplace operators). From now on, we assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M45">View MathML</a>, which is without any loss of generality as can be seen from assumptions (A)(ii), (iii).

In particular, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M46">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M47">View MathML</a> stands for the usual p-Laplacian <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M48">View MathML</a>, we can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M49">View MathML</a> in (A). Conversely, in the case where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M49">View MathML</a> holds in (A), by the inequalities in Remark 3(ii) and (iii), we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M51">View MathML</a> whence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M46">View MathML</a>. Hence, our equation contains the p-Laplace equation as a special case.

In the case where f does not depend on the gradient of u, there are many existence results because our equation has the variational structure (cf.[1,4,8]). Although there are a few results for our equation (P) with f including ∇u, we can refer to [7,9] and [10] for the existence of a positive solution in the case of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M53">View MathML</a>-Laplacian or m-Laplacian (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M54">View MathML</a>). In particular, in [9] and [7], the nonlinear term f is imposed to be nonnegative. The results in [7] and [10] are applied to the m-Laplace equation with an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M55">View MathML</a>-superlinear term f w.r.t. u. Here, we mention the result in [9] for the p-Laplacian. Faria, Miyagaki and Motreanu considered the case where f is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M8">View MathML</a>-sublinear w.r.t. u and ∇u, and they supposed that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M57">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M58">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M59">View MathML</a>. The purpose of this paper is to remove the sign condition and to admit the condition like <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M60">View MathML</a> for large <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M61">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M62">View MathML</a>. Concerning the condition for f as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M63">View MathML</a>, Zou in [10] imposed that there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M64">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M65">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M66">View MathML</a> for the m-Laplace problem. Hence, we cannot apply the result of [10] and [9] to the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M67">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M62">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M69">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M70">View MathML</a> (admitting sign changes), but we can do our result if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M61">View MathML</a> is large.

In [9], the positivity of a solution is proved by the comparison principle. However, since we are not able to do it for our operator in general, after we provide a non-negative and non-trivial solution as a limit of positive approximate solutions (in Section 2), we obtain the positivity of it due to the strong maximum principle for our operator.

1.1 Statements

To state our first result, we define a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M72">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M73">View MathML</a>

(2)

which is equal to 1 in the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M46">View MathML</a> (i.e., the case of the p-Laplacian) because we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M49">View MathML</a>. Then, we introduce the hypothesis (f1) to the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M76">View MathML</a> in (f) as t is small.

(f1) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M70">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M78">View MathML</a> such that the Lebesgue measure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M79">View MathML</a> is positive and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M80">View MathML</a>

(3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M38">View MathML</a> is the continuous function in (f) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M82">View MathML</a> is the first positive eigenvalue of the p-Laplacian with the weight function m obtained by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M83">View MathML</a>

(4)

Theorem 1Assume (f1). Then equation (P) has a positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M84">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M85">View MathML</a>

andνdenotes the outward unit normal vector onΩ.

Next, we consider the case where A is asymptotically <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M8">View MathML</a>-homogeneous near zero in the following sense:

(AH0) There exist a positive function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M87">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M88">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M89">View MathML</a>

(5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M90">View MathML</a>

(6)

Under (AH0), we can replace the hypothesis (f1) with the following (f2):

(f2) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M70">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M92">View MathML</a> such that (3) and the Lebesgue measure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M79">View MathML</a> is positive, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M94">View MathML</a> is the first positive eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M95">View MathML</a> with a weight function m obtained by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M96">View MathML</a>

(7)

Theorem 2Assume (AH0) and (f2). Then equation (P) has a positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M84">View MathML</a>.

Throughout this paper, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M98">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M100">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M25">View MathML</a> because we consider the existence of a positive solution only. In what follows, the norm on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M102">View MathML</a> is given by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M103">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M104">View MathML</a> denotes the usual norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M105">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M106">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M107">View MathML</a>). Moreover, we denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M108">View MathML</a>.

1.2 Properties of the map A

Remark 3 The following assertions hold under condition (A):

(i) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M110">View MathML</a> is maximal monotone and strictly monotone in y;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M111">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M112">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M113">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M112">View MathML</a>,

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M13">View MathML</a> are the positive constants in (A).

Proposition 4 ([[3], Proposition 1])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M117">View MathML</a>be a map defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M118">View MathML</a>

for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M119">View MathML</a>. ThenAis maximal monotone, strictly monotone and has<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M120">View MathML</a>property, that is, any sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M121">View MathML</a>weakly convergent touwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M122">View MathML</a>strongly converges tou.

2 Constructing approximate solutions

Choose a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M123">View MathML</a>. In this section, for such ψ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M124">View MathML</a>, we consider the following elliptic equation:

In [7], the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M126">View MathML</a> in the above equation is considered.

Lemma 5Suppose (f1) or (f2). Then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M127">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M128">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M130">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M25">View MathML</a>.

Proof From the growth condition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M38">View MathML</a> and (3), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M133">View MathML</a>

holds, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M134">View MathML</a> is a positive constant independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M135">View MathML</a>. Therefore, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M136">View MathML</a>, we easily see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M137">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M130">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M25">View MathML</a> holds. □

Proposition 6If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M141">View MathML</a>is a non-negative solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M142">View MathML</a>) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M143">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M144">View MathML</a>. Moreover, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M145">View MathML</a>, there exists a positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M146">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M147">View MathML</a>holds for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M148">View MathML</a>.

Proof Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M149">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M150">View MathML</a>, and in the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M152">View MathML</a> is an arbitrarily fixed constant. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M153">View MathML</a> be a non-negative solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M142">View MathML</a>) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M155">View MathML</a> (some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M145">View MathML</a>). For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M157">View MathML</a>, choose a smooth increasing function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M158">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M159">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M160">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M161">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M162">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M163">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M164">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M165">View MathML</a>. Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M166">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M167">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M168">View MathML</a>, then by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M169">View MathML</a> as a test function (note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M170">View MathML</a> is bounded), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M171">View MathML</a>

(8)

due to Remark 3(iii) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M172">View MathML</a>. Putting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M173">View MathML</a>, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M174">View MathML</a> provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M175">View MathML</a> (note <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M157">View MathML</a>). Similarly, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M177">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M178">View MathML</a>, and if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M179">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M180">View MathML</a> (note <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M181">View MathML</a>). Thus, according to Young’s inequality, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M182">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M183">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M184">View MathML</a>

(9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M173">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M186">View MathML</a> (>1). As a result, because of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M187">View MathML</a>, according to Hölder’s inequality and the monotonicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M188">View MathML</a> with respect to r on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M189">View MathML</a>, taking a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M190">View MathML</a> and setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M191">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M192">View MathML</a>

(10)

provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M168">View MathML</a> by (8) and (9), where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M194">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M195">View MathML</a> comes from the continuous embedding of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M196">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M197">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M198">View MathML</a> is a positive constant independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M153">View MathML</a>, ε and r. Consequently, Moser’s iteration process implies our conclusion. In fact, we define a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M200">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M201">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M202">View MathML</a>. Then, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M203">View MathML</a> holds if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M204">View MathML</a> by applying Fatou’s lemma to (10) and letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M205">View MathML</a>. Here, we also see <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M206">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M207">View MathML</a>. Therefore, by the same argument as in Theorem C in [4], we can obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M144">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M147">View MathML</a> for some positive constant D independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M153">View MathML</a> and ε. □

Lemma 7Suppose (f1) or (f2). If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M141">View MathML</a>is a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M142">View MathML</a>) for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M124">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M214">View MathML</a>.

Proof Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M215">View MathML</a> as a test function in (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M142">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M217">View MathML</a>

because of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M98">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M99">View MathML</a> and by Remark 3(iii). Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M220">View MathML</a> follows. Because Proposition 6 guarantees that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M144">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M222">View MathML</a> (for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M223">View MathML</a>) by the regularity result in [11]. Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M224">View MathML</a> because of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M124">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M226">View MathML</a>. In addition, Lemma 5 implies the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M127">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M228">View MathML</a> in the distribution sense. Therefore, according to Theorem A and Theorem B in [4], <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M229">View MathML</a> in Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M230">View MathML</a> on Ω, namely, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M214">View MathML</a>. □

The following result can be shown by the same argument as in [[9], Theorem 3.1].

Proposition 8Suppose (f1) or (f2). Then, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M124">View MathML</a>, (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M142">View MathML</a>) has a positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M214">View MathML</a>.

Proof Fix any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M124">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M236">View MathML</a> be a Schauder basis of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M102">View MathML</a> (refer to [12] for the existence). For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M238">View MathML</a>, we define the m-dimensional subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M239">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M102">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M241">View MathML</a>. Moreover, set a linear isomorphism <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M242">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M243">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M244">View MathML</a> be a dual map of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M245">View MathML</a>. By identifying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M246">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M247">View MathML</a>, we may consider that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M248">View MathML</a> maps from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M249">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M246">View MathML</a>. Define maps <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M251">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M252">View MathML</a> from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M239">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M249">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M255">View MathML</a>

for u, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M256">View MathML</a>. We claim that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M238">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M258">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M259">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M249">View MathML</a>. Indeed, by the growth condition of f, Remark 3(iii) and Hölder’s inequality, we easily have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M261">View MathML</a>

(11)

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M262">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M263">View MathML</a>. This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M264">View MathML</a> is coercive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M239">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M266">View MathML</a>. Set a homotopy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M267">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M268">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M269">View MathML</a>. By recalling that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M264">View MathML</a> is coercive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M239">View MathML</a>, we see that there exists an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M272">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M273">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M268">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M275">View MathML</a> because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M276">View MathML</a> and the norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M246">View MathML</a> are equivalent on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M239">View MathML</a>. Therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M279">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M280">View MathML</a> is the identity map on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M246">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M282">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M283">View MathML</a> denotes the degree on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M246">View MathML</a> for a continuous map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M285">View MathML</a> (cf.[13]). Hence, this yields the existence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M286">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M287">View MathML</a>, and so the desired <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M288">View MathML</a> is obtained by setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M289">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M248">View MathML</a> is injective.

Because (11) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M291">View MathML</a> leads to the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M292">View MathML</a> by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M266">View MathML</a>, we may assume, by choosing a subsequence, that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M288">View MathML</a> converges to some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M295">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M102">View MathML</a> and strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M297">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M298">View MathML</a> be a natural projection onto <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M239">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M300">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M301">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M302">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M259">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M249">View MathML</a>, by noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M305">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M239">View MathML</a> for a map A defined in Proposition 4, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M307">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M207">View MathML</a>, where we use the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M292">View MathML</a>, the growth condition of f and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M310">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M297">View MathML</a>. In addition, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M312">View MathML</a> is bounded, by the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M292">View MathML</a>, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M314">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M207">View MathML</a>, whence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M316">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M207">View MathML</a> holds. As a result, it follows from the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M120">View MathML</a> property of A that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M310">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M102">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M207">View MathML</a>.

Finally, we shall prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M295">View MathML</a> is a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M142">View MathML</a>). Fix any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M324">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M325">View MathML</a>. For each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M326">View MathML</a>, by letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M207">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M328">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M329">View MathML</a>

(12)

Since l is arbitrary, (12) holds for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M330">View MathML</a>. Moreover, the density of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M331','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M331">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M102">View MathML</a> guarantees that (12) holds for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M44">View MathML</a>. This means that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M295">View MathML</a> is a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M142">View MathML</a>). Consequently, our conclusion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M336">View MathML</a> follows from Lemma 7. □

3 Proof of theorems

Lemma 9Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M337">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M338">View MathML</a>

holds, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M72">View MathML</a>is the positive constant defined by (2).

Proof Because of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M337">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M341">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M342">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M343">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M344">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M345">View MathML</a> in Ω. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M346">View MathML</a> hold. Therefore, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M347">View MathML</a>

(13)

in Ω by (ii) and (iii) in Remark 3 and Young’s inequality. □

Lemma 10Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M348">View MathML</a>and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M337">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M350">View MathML</a>

holds.

Proof First, we note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M346">View MathML</a> hold by the same reason as in Lemma 9. Applying Young’s inequality to the second term of the right-hand side in (14) (refer to (13) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M49">View MathML</a>), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M353">View MathML</a>

(14)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M354">View MathML</a>

(15)

in Ω. Similarly, we also have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M355">View MathML</a>

(16)

The conclusion follows from (15) and (16). □

Under (f1) or (f2), we denote a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M214">View MathML</a> of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M142">View MathML</a>) for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M124">View MathML</a> obtained by Proposition 8.

Lemma 11Assume (f1) or (f2). Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M359">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M360">View MathML</a>is bounded in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M102">View MathML</a>.

Proof Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M153">View MathML</a> as a test function in (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M142">View MathML</a>), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M364">View MathML</a>

by Remark 3(iii), the growth condition of f, Hölder’s inequality and the continuity of the embedding of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M102">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M297">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M367">View MathML</a> (<p) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M134">View MathML</a> is a positive constant independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M153">View MathML</a>. Because of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M266">View MathML</a>, this yields the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M371">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M372">View MathML</a>). □

Lemma 12Assume (f1) or (f2). Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M373">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M374','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M374">View MathML</a>hold for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M124">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M376','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M376">View MathML</a>denotes the Lebesgue measure of Ω, and where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M12">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M378">View MathML</a>are positive constants as in (A) and Lemma 5, respectively.

Proof Fix any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M124">View MathML</a> and choose any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M380">View MathML</a>. By taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M381">View MathML</a> as a test function, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M382">View MathML</a>

(17)

by Lemma 5 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M383">View MathML</a>. On the other hand, by Remark 3(iii) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M384">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M385">View MathML</a>

(18)

Therefore, (17) and (18) imply the inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M386">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M380">View MathML</a>. As a result, by letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M388">View MathML</a>, our conclusion is shown. □

Lemma 13Assume (f2) and (AH0). Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M389">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M390">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M391">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M392">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M393">View MathML</a>

holds, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M394">View MathML</a>is a continuous function as in (AH0).

Proof Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M395">View MathML</a> hold (as in the proof of Lemma 9). Because we easily see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M396">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M42">View MathML</a> with some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M398">View MathML</a> independent of u (see (6)), it is sufficient to show <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M399">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M392">View MathML</a>. Here, we fix any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M182">View MathML</a>. By the property of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M394">View MathML</a> (see (6)) and because we are assuming that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M390','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M390">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M391">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M392">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M406','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M406">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M100">View MathML</a> provided sufficiently small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M124">View MathML</a>. Therefore, for such sufficiently small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M124">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M410">View MathML</a>

because of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M373">View MathML</a> by Lemma 12. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M182">View MathML</a> is arbitrary, our conclusion is shown. □

3.1 Proof of main results

Proof of Theorems

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M413">View MathML</a>. Due to Proposition 6 and Lemma 11, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M414">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M415">View MathML</a> independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M413">View MathML</a>. Hence, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M417','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M417">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M223">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M419','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M419">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M420','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M420">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M413">View MathML</a> by the regularity result in [11]. Because the embedding of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M422','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M422">View MathML</a> into <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M391">View MathML</a> is compact and by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M214">View MathML</a>, there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M425','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M425">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M426','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M426">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M427','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M427">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M428','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M428">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M391">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M430">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M431">View MathML</a> occurs, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M336">View MathML</a> by the same reason as in Lemma 7, and hence our conclusion is proved. Now, we shall prove <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M431','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M431">View MathML</a> by contradiction for each theorem. So, we suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M434','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M434">View MathML</a>, whence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M435">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M391">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M430">View MathML</a>.

Proof of Theorem 1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M389','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M389">View MathML</a> be an eigenfunction corresponding to the first positive eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M82">View MathML</a> (cf.[14,15], it is well known that we can obtain φ as the minimizer of (4)), namely, φ is a positive solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M440','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M440">View MathML</a> in Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M441">View MathML</a> on Ω. Since p-Laplacian is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M8">View MathML</a>-homogeneous, we may assume that φ satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M443','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M443">View MathML</a>, and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M444','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M444">View MathML</a> holds by taking φ as a test function. Choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M380">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M446','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M446">View MathML</a> (note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M447','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M447">View MathML</a> as in (f1)). Due to (f1), there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M182">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M449','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M449">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M450','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M450">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M100">View MathML</a>. Since we are assuming <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M435">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M391">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M430">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M455','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M455">View MathML</a> occurs for sufficiently large n. Then, for such sufficiently large n, according to Lemma 9, (1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M456">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M457','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M457">View MathML</a>

This is a contradiction. □

Proof of Theorem 2 Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M458','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M458">View MathML</a> holds, by the standard argument as in the p-Laplacian, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M459','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M459">View MathML</a> and it is the first positive eigenvalue of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M460','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M460">View MathML</a> in Ω and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M441','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M441">View MathML</a> on Ω. Therefore, by the well-known argument, there exists a positive eigenfunction <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M462">View MathML</a> corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M94">View MathML</a> (we can obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M464">View MathML</a> as the minimizer of (7)). Hence, by taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M464','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M464">View MathML</a> as a test function, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M466','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M466">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M467','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M467">View MathML</a> follows. Because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M468','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M468">View MathML</a> is a solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M469','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M469">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M462','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M462">View MathML</a> is an eigenfunction corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M94">View MathML</a>, according to Lemma 11 and Lemma 13 (note <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M472','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M472">View MathML</a> as in (AH0)), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M473','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M473">View MathML</a>

(19)

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M430','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M430">View MathML</a> since we are assuming <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M435','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M435">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M391">View MathML</a>, where we use the facts that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M456','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M456">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M478','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M478">View MathML</a> in Ω. Furthermore, by Fatou’s lemma and (3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M479','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M479">View MathML</a>

As a result, by taking a limit superior with respect to n in (19), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M480','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/173/mathml/M480">View MathML</a>. This is a contradiction. □

Competing interests

The author declares that she has no competing interests.

Acknowledgements

The author would like to express her sincere thanks to Professor Shizuo Miyajima for helpful comments and encouragement. The author thanks Professor Dumitru Motreanu for giving her the opportunity of this work. The author thanks referees for their helpful comments.

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