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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 periodic solutions of a p-Laplacian-Neumann problem

Raad Awad Hameed12, Jiebao Sun1 and Boying Wu1*

Author Affiliations

1 Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, P.R. China

2 Department of Mathematics, College of Education, Tikrit University, Tikrit, Iraq

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Boundary Value Problems 2013, 2013:171  doi:10.1186/1687-2770-2013-171


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/171


Received:6 March 2013
Accepted:7 June 2013
Published:23 July 2013

© 2013 Hameed et al.; licensee Springer

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

Abstract

In this paper, we study a periodic p-Laplacian equation with nonlocal terms and Neumann boundary conditions. We establish the existence of time periodic solutions of the p-Laplacian-Neumann problem by the theory of Leray-Schauder degree.

1 Introduction

In this paper, we consider the periodic boundary problem for a p-Laplacian equation of the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M1">View MathML</a>

(1.1)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M2">View MathML</a>

(1.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M3">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M4">View MathML</a>, Ω is a bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M5">View MathML</a> with smooth boundary Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M6">View MathML</a> denotes the outward normal derivative on Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M7">View MathML</a>. This problem is motivated by models which have been proposed for some problems in mathematical biology. The function u represents the spatial densities of the species at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M8">View MathML</a>; the diffusion term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M9">View MathML</a> represents the effect of dispersion in the habitat, which models a tendency to avoid crowding, and the speed of the diffusion is slow since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M4">View MathML</a>; the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M11">View MathML</a> models the contribution of the population supply due to births and deaths; the Neumann boundary conditions model the trend of the biology population to survive on the boundary. Assumptions of m, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M12">View MathML</a> will be introduced in the next section.

The model as (1.1) was first studied by Allegretto and Nistri. In [1] they studied the existence of nontrivial nonnegative periodic solutions and optimal control for the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M13">View MathML</a>

with Dirichlet boundary value conditions. Later, many mathematical researchers studied extended forms of this kind of equation. For example, in [2-7], the authors considered some nonlinear diffusion equations with nonlocal terms such as the porous equation with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M14">View MathML</a>, the p-Laplacian equation with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M4">View MathML</a> and the doubly degenerate parabolic equation. All these problems are the Dirichlet boundary value conditions, and these boundary conditions describe that the boundary we consider in this model is lethal to the species. Moreover, the methods in these papers are all based on the theory of Leray-Schauder degree. However, there are few results on degenerate periodic parabolic equations with nonlocal terms and Neumann boundary conditions. Recently, in [8], Wang and Yin considered the following periodic Neumann boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M16">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M14">View MathML</a>. By the parabolic regularized method and the theory of Leray-Schauder degree, they established the existence of nontrivial nonnegative periodic solutions. Also, there are many works about reaction diffusion equations without nonlocal term; one can see [9-13] and the references therein, and the boundary value condition and research method are different from our work.

In this paper, we consider the periodic solution of p-Laplacian Neumann problem (1.1)-(1.3). In [14], the authors studied equation (1.1) with the Dirichlet boundary value condition. Compared with the Dirichlet boundary value condition in [14], the Neumann boundary value condition causes an additional difficulty in establishing some a priori estimates. On the other hand, different from that in the case of the Dirichlet boundary value condition, the standard regularized problem of problem (1.1)-(1.3) is not well posed, and thus a modified regularized problem for (1.1)-(1.3) is considered. In addition, we will make use of the Moser iterative method to establish the a priori upper bound of the solution of the regularized problem. By the theory of Leray-Schauder degree, we prove that this modified problem admits nontrivial nonnegative periodic solutions. Then, by passing to a limit process, we obtain the existence of nontrivial nonnegative periodic solutions of problem (1.1)-(1.3). In the process of proving the main results, the nonlocal term, which reflects the reality of the model (1.1), will cause a difficulty when we establish a lower bound estimate of the maximum modulus of the solution of the regularized problem. Otherwise, we can use the method of upper and lower solution to prove the existence of periodic solutions. At last, the existence theorem shows that the spatial densities of the species are periodic under the case of nonlinear diffusion.

This paper is organized as follows. In Section 2, we show some necessary preliminaries including the modified regularized problem. In Section 3, we establish some necessary a priori estimations of the solution of the modified regularized problem. Then we obtain the main results of this paper.

2 Preliminaries

In this paper, we assume that

(A1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M18">View MathML</a> is a bounded continuous functional satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M19">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M20">View MathML</a> are constants independent of u, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M22">View MathML</a>;

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M23">View MathML</a> and satisfies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M24">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M25">View MathML</a> denotes the set of functions which are continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M26">View MathML</a> and of T-periodic with respect to t.

From (A2), we can see that there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M29">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M30">View MathML</a>

Since equation (1.1) is degenerate at points where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M31">View MathML</a>, problem (1.1)-(1.3) has no classical solutions in general, so we focus on the discussion of weak solutions in the sense of the following.

Definition 1 A function u is said to be a weak solution of problem (1.1)-(1.3) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M32">View MathML</a> and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M33">View MathML</a>

(2.1)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M34">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M35">View MathML</a>.

Due to the degeneracy of equation (1.1), we consider the following regularized problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M36">View MathML</a>

(2.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M37">View MathML</a>

(2.3)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M38">View MathML</a>

(2.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M39">View MathML</a> and ε is a sufficiently small positive constant. The desired solution will be obtained as the limit point of the solutions of problem (2.2)-(2.4). In the following, we introduce a map by the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M40">View MathML</a>

(2.5)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M41">View MathML</a>

(2.6)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M42">View MathML</a>

(2.7)

Then we can define a map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M43">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M44">View MathML</a>. Applying classical estimates (see [15]), we can know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M45">View MathML</a> is bounded by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M46">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M47">View MathML</a> is Hölder continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M48">View MathML</a>. Then by the Arzela-Ascoli theorem, the map G is compact. So, the map is a compact continuous map. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M49">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M50">View MathML</a>, we can see that the nonnegative solution of problem (2.2)-(2.4) is also a nonnegative solution solving <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M51">View MathML</a>. So, we will study the existence of nonnegative fixed points of the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M51">View MathML</a> instead of the nonnegative solutions of problem (2.2)-(2.4).

3 Proof of the main results

First, by the same method as in [14], we can obtain the nonnegativity of the solution of problem (2.2)-(2.4).

Lemma 1If a nontrivial function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M53">View MathML</a>solves<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M51">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M55">View MathML</a>

In the following, by the Moser iterative technique, we will show the a priori estimate for the upper bound of nonnegative periodic solutions of problem (2.5)-(2.7). Here and below we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M56">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M57">View MathML</a>) the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M58">View MathML</a> norm.

Lemma 2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M47">View MathML</a>be a nonnegative periodic solution solving<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M61">View MathML</a>, then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M62">View MathML</a>independent ofλ, εsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M63">View MathML</a>

(3.1)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M64">View MathML</a>.

Proof Multiplying Eq. (2.5) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M65">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M66">View MathML</a>) and integrating over Ω, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M67">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M68">View MathML</a>

(3.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M69">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M70">View MathML</a>) are positive constants independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M47">View MathML</a> and m.

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M72">View MathML</a> and set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M73">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M75">View MathML</a>. For convenience, we denote by C a positive constant independent of k and m, which may take different values. From (3.2) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M76">View MathML</a>

(3.3)

By using the Gagliardo-Nirenberg inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M77">View MathML</a>

(3.4)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M78">View MathML</a>

By inequalities (3.3), (3.4) and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M79">View MathML</a>, we obtain the following differential inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M80">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M81">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M82">View MathML</a>

(3.5)

For Young’s inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M83">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M84">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M88">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M89">View MathML</a>, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M90">View MathML</a>

then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M91">View MathML</a>

(3.6)

Here we have used the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M92">View MathML</a> for some r independent of k. In fact, it is easy to verify that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M93">View MathML</a>

Denoting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M94">View MathML</a>

and combining (3.6), (3.5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M95">View MathML</a>

(3.7)

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M96">View MathML</a>

(3.8)

From the periodicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M97">View MathML</a>, we know that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M98">View MathML</a> at which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M99">View MathML</a> reaches its maximum and thus the left-hand side of (3.8) vanishes. Then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M100">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M101">View MathML</a>

Therefore we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M102">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M103">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M104">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M105">View MathML</a> are bounded, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M106">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M107">View MathML</a> is a positive constant independent of k. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M108">View MathML</a> implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M110">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M111">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M112">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M113">View MathML</a>. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M114">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M115">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M116">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M117">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M118">View MathML</a>

(3.9)

On the other hand, it follows from (3.2) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M119">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M120">View MathML</a>

(3.10)

By Hölder’s inequality and Sobolev’s theorem, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M121">View MathML</a>

(3.11)

Combined with (3.10), it yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M122">View MathML</a>

By Young’s inequality, it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M123">View MathML</a>

(3.12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M69">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M70">View MathML</a>) are constants independent of u. Taking the periodicity of u into account, we infer from (3.12) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M126">View MathML</a>

which together with (3.9) implies (3.1). The proof is completed. □

Corollary 1There exists a positive constantRindependent ofεsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M127">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M128">View MathML</a>is a ball centered at the origin with radiusRin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M129">View MathML</a>.

Proof It follows from Lemma 2 that there exists a positive constant R independent of ε such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M130">View MathML</a>

Hence the degree is well defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M128">View MathML</a>. From the homotopy invariance of the Leray-Schauder degree, we can see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M132">View MathML</a>

(3.13)

From the existence and uniqueness of the solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M133">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M134">View MathML</a>. That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M135">View MathML</a>. The proof is completed. □

Lemma 3There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M136">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M137">View MathML</a>such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M140">View MathML</a>admits no nontrivial solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M47">View MathML</a>satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M142">View MathML</a>

whereris a positive constant independent ofε.

Proof By contradiction, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M47">View MathML</a> be a nontrivial solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M51">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M145">View MathML</a>. For any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M146">View MathML</a>, multiplying (2.5) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M147">View MathML</a> and integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M148">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M149">View MathML</a>

(3.14)

By the periodicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M47">View MathML</a>, the first term on the left-hand side in (3.14) is zero. As in the proof of Lemma 2.2 of [14], the second term on the left-hand side in (3.14) can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M151">View MathML</a>

and thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M152">View MathML</a>

(3.15)

Combining (3.15) with (3.14), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M153">View MathML</a>

(3.16)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M154">View MathML</a> be the first eigenvalue of the p-Laplacian equation on Ω with zero boundary conditions and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M155">View MathML</a> be the corresponding eigenfunction. We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M156">View MathML</a>

And also we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M157">View MathML</a> can be strictly positive in the subfield <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M158">View MathML</a>. Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M159">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M160">View MathML</a>

(3.17)

Multiplying (2.5) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M47">View MathML</a> and integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M162">View MathML</a>, from the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M145">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M164">View MathML</a>

(3.18)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M165">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M166">View MathML</a> denotes the Lebesgue measure of the domain Ω. Combining (3.18) with (3.17), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M167">View MathML</a>

(3.19)

In addition, the assumptions (A1), (A2) give

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M168">View MathML</a>

(3.20)

The above two inequalities imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M169">View MathML</a>

(3.21)

Obviously, we can choose suitably small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M170">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M171">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M172">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M173">View MathML</a>, the inequality (3.21) does not hold. It is a contradiction. The proof is completed. □

Corollary 2There exists a small positive constantrwhich is independent ofεand satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M174">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M175">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M176">View MathML</a>is a ball centered at the origin with radiusrin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M129">View MathML</a>.

Proof Similar to Lemma 3, we can see that there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M178">View MathML</a> independent of ε such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M179">View MathML</a>

Hence the degree is well defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M176">View MathML</a>. From the homotopy invariance of the Leray-Schauder degree, we can see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M181">View MathML</a>

(3.22)

Lemma 3 shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M182">View MathML</a> admits no nontrivial solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M176">View MathML</a> and it is also easy to see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M184">View MathML</a> is not a solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M182">View MathML</a>. So, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M186">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M187">View MathML</a>

The proof is completed. □

Theorem 1If assumptions (A1) and (A2) hold, then problem (1.1)-(1.3) admits a nontrivial nonnegative periodic solutionu.

Proof Using Corollary 1 and Corollary 2, we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M188">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M189">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M190">View MathML</a> is a ball centered at the origin with radius ξ in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M129">View MathML</a>, R and r are positive constants and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M192">View MathML</a>. By the theory of the Leray-Schauder degree and Lemma 1, we can conclude that problem (2.2)-(2.4) admits a nontrivial nonnegative periodic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M47">View MathML</a>. By Lemma 3 and a similar method to that in [14], we can obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M194">View MathML</a>

Combining with the regularity results [16] a similar argument to that in [17], we can prove that the limit function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/171/mathml/M47">View MathML</a> is a nonnegative nontrivial periodic solution of problem (1.1)-(1.3). □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

RH, JS and BW carried out the proof of the main part of this article, BW corrected the manuscript and participated in its design and coordination. All authors have read and approved the final manuscript.

Acknowledgements

This work is partially supported by the National Science Foundation of China (11271100, 11126222), the Fundamental Research Funds for the Central Universities (Grant No. HIT. NSRIF. 2011006), the Natural Sciences Foundation of Heilongjiang Province (QC2011C020) and also the 985 project of Harbin Institute of Technology.

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