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This article is part of the series Jean Mawhin’s Achievements in Nonlinear Analysis.

Open Access Research

Periodic solution of a quasilinear parabolic equation with nonlocal terms and Neumann boundary conditions

Raad Awad Hameed12, Boying Wu1 and Jiebao Sun1*

Author Affiliations

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

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

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Boundary Value Problems 2013, 2013:34  doi:10.1186/1687-2770-2013-34

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/34


Received:26 November 2012
Accepted:2 February 2013
Published:21 February 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 article, we study the periodic solution of a quasilinear parabolic equation with nonlocal terms and Neumann boundary conditions. By using the theory of Leray-Schauder degree, we obtain the existence of a nontrivial nonnegative time periodic solution.

1 Introduction

The aim of this work is to consider the following periodic problem for a quasilinear parabolic equation:

(1.1)

(1.2)

(1.3)

where Ω is a bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M4">View MathML</a> with smooth boundary Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M5">View MathML</a> denotes the outward normal derivative on Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M7">View MathML</a> satisfies some suitable smoothness and structure conditions. This model can be used to describe the models for some interesting phenomena in mathematical biology, fisheries and wildlife management. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M8">View MathML</a> gives the number of individuals (per unit area) of the species at position x and time t, where x represents the spatial variable and t represents the time. The term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M9">View MathML</a> models a tendency to avoid high density in the habitat, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M10">View MathML</a> describes the ways in which a given population grows and shrinks over time, as controlled by birth, death, emigration or immigration, and the Neumann boundary condition models the trend of the biology population who survive on the boundary.

In last decades, linear parabolic equations with nonlocal terms have been investigated by numerous researchers [1-4]. A typical model was submitted by Allegretto and Nistri [1] and they proposed the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M11">View MathML</a>

with the Dirichlet boundary conditions. Also, according to the actual needs, many authors divert attention to nonlinear diffusion equations with nonlocal terms such as the porous equation [5,6] with a typical form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M12">View MathML</a>

(1.4)

and the p-Laplacian equation [7] with a typical form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M13">View MathML</a>

(1.5)

Equation (1.4) is degenerate if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M14">View MathML</a> and singular if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M15">View MathML</a>. Equation (1.5) is degenerate if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M16">View MathML</a> and singular if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M17">View MathML</a>. Only the cases <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M14">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M16">View MathML</a> are considered with a few exceptions. All these equations are considered with the Dirichlet boundary condition which describes that the boundary is lethal to the species. Moreover, the methods in these papers are all based on the theory of Leray-Schauder degree. However, results on the quasilinear periodic parabolic equations with nonlocal terms and Neumann boundary conditions are few. In a recent paper [8], Wang and Yin considered the following periodic Neumann boundary value problem:

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M21">View MathML</a>. By the parabolic regularized method and the theory of Leray-Schauder degree, they established the existence of nontrivial nonnegative periodic solutions.

Inspired by the work of [8], we consider the periodic solutions of the Neumann boundary value problem of a quasilinear parabolic equation with nonlocal terms. Compared with the Dirichlet boundary condition, the Neumann boundary condition causes an additional difficulty in establishing some a priori estimates. On the other hand, different from the cases of the Dirichlet boundary condition, an auxiliary problem for (1.1)-(1.3) is considered for using the theory of Leray-Schauder degree. We prove that this problem (1.1)-(1.3) admits a nontrivial nonnegative periodic solution, that is, the following theorem.

Theorem 1If assumptions (A1), (A2), (A3) hold, then problem (1.1)-(3.3) admits a nontrivial nonnegative periodic solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M22">View MathML</a>.

The article is organized in the following way. In Section 2, we give some necessary preliminaries including the auxiliary problem. In Section 3, we establish some necessary a priori estimations of the solutions of the auxiliary problem. Then we show the proof of the main result of this paper.

2 Preliminaries

In the paper, we assume that

(A1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M23">View MathML</a> and there exist two constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M24">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M25">View MathML</a>

where is the class of functions which are continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M27">View MathML</a> and T-periodic with respect to t. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M28">View MathML</a> is continuous with respect to u.

(A2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M29">View MathML</a> is a bounded continuous functional satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M30">View MathML</a>

where C is a positive constant independent of u, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M32">View MathML</a>.

(A3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M33">View MathML</a> and satisfies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M34">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M35">View MathML</a> is the first eigenvalue of the Laplacian equation on ω with zero boundary and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M36">View MathML</a> is the corresponding eigenfunction.

Since the regularity follows from a quite standard approach, we focus on the discussion of weak solutions in the following sense.

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/2013/1/34/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M37">View MathML</a> and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M38">View MathML</a>

(2.1)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M39">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M40">View MathML</a>.

In order to use the theory of Leray-Schauder degree, we introduce a map by considering the following auxiliary problem:

(2.2)

(2.3)

(2.4)

where ε is a sufficiently small positive constant, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M44">View MathML</a> is a parameter and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M45">View MathML</a>. Then we can define a map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M46">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M47">View MathML</a>. Applying classical estimates (see [9]), we can see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M48">View MathML</a> is bounded by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M49">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M50">View MathML</a> is Hölder continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M51">View MathML</a>. Then, by the Arzela-Ascoli theorem, the map G is compact. So, the map G is a compact continuous map. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M52">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M53">View MathML</a>, we can see that the nonnegative solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M50">View MathML</a> of problem (2.2)-(2.4) is also a nonnegative fixed point of the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M55">View MathML</a>. So, we will study the existence of nonnegative fixed points of the map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M56">View MathML</a> instead of a nonnegative solution of problem (2.2)-(2.4). And the desired solution u of (1.1)-(1.3) would be obtained as a limit point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M50">View MathML</a>.

3 The proof of the main result

First, by the same method as in [4], we can obtain the nonnegativity of the solutions of problem (2.2)-(2.4).

Lemma 1If a nontrivial function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M58">View MathML</a>solves<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M55">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M60">View MathML</a>

In the following, we will show some a priori estimates for the upper bound of a nonnegative periodic solution of problem (2.2)-(2.4). Here and below, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M61">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M62">View MathML</a>) the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M63">View MathML</a> norm.

Lemma 2For<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M64">View MathML</a>, let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M65">View MathML</a>be a nonnegative periodic solution which solves<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M66">View MathML</a>, then there exists a constantKindependent ofλ, εsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M67">View MathML</a>

(3.1)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M68">View MathML</a>.

Proof Multiplying (2.2) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M69">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M70">View MathML</a>) and integrating over Ω, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M71">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M72">View MathML</a>

(3.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M73">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M74">View MathML</a>) are positive constants independent of u and m. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M75">View MathML</a> and set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M76">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M77">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/2013/1/34/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M78">View MathML</a>

(3.3)

By using the Gagliardo-Nirenberg inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M79">View MathML</a>

(3.4)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M80">View MathML</a>

By inequalities (3.3), (3.4) and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M81">View MathML</a>, we obtain the following differential inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M82">View MathML</a>

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M83">View MathML</a>

we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M84">View MathML</a>

(3.5)

By Young’s inequality,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M85">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M90">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M91">View MathML</a>. Set

then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M93">View MathML</a>

(3.6)

Here we have used the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M94">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/2013/1/34/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M95">View MathML</a>

Denoting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M96">View MathML</a>

and combining (3.5) with (3.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M97">View MathML</a>

(3.7)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M98">View MathML</a>

(3.8)

From the periodicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M99">View MathML</a>, we know that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M100">View MathML</a> at which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M101">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/2013/1/34/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M102">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M103">View MathML</a>

Therefore, we conclude that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M104">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M105">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M106">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M107">View MathML</a> are bounded, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M108">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M109">View MathML</a> is a positive constant independent of k. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M110">View MathML</a>

thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M111">View MathML</a>

or

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M112">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M113">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M114">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M115">View MathML</a>

(3.9)

Now, we just need to show the estimate of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M116">View MathML</a>. Multiplying (2.2) by u and integrating by parts over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M51">View MathML</a>, by the periodicity of u, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M118">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M119">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M120">View MathML</a>, by assumption (A2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M121">View MathML</a>

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M122">View MathML</a>

where C is a positive independent of λ. By Young’s inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M123">View MathML</a>

Combining with the above inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M124">View MathML</a>

which together with (3.9) implies (3.1), and thus the proof is complete. □

Corollary 1There exists a positive constant R independent ofεsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M125">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M126">View MathML</a>is a ball centered at the origin with radiusRin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M127">View MathML</a>.

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M128">View MathML</a>

So, the degree is well defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M126">View MathML</a>. From the homotopy invariance of the Leray-Schauder degree and the existence and uniqueness of the solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M130">View MathML</a>, we can see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M131">View MathML</a>

The proof is completed. □

Lemma 3There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M132">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M133">View MathML</a>such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M135">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M136">View MathML</a>admits no nontrivial solutionusatisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M137">View MathML</a>

whereris a positive constant independent ofε.

Proof By contradiction, let u be a nontrivial fixed point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M138">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M139">View MathML</a>. For any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M140">View MathML</a>, multiplying (2.2) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M141">View MathML</a> and integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M142">View MathML</a>, we have

(3.10)

By the periodicity of u, the first term on the left-hand side is zero. The second term on the left-hand side can be rewritten as

The third term of the left-hand side of equation (3.11) can be rewritten as

Then from (3.10), we obtain

From assumption (A1), we can see that

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M44">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M149">View MathML</a>

By an approaching process, we choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M150">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M151">View MathML</a> is the eigenvector of the first eigenvalue <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M35">View MathML</a> in (A3), and then we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M153">View MathML</a>

(3.11)

Thus, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M154">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M155">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M156">View MathML</a>

From assumption (A2), we can see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M157">View MathML</a>

holds for any sufficiently small r and ε, which is a contradiction to assumption (A3). The proof is complete. □

Corollary 2There exists a small positive constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M158">View MathML</a>which is independent ofε, τsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M159">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M160">View MathML</a>is a ball centered at the origin with radiusrin<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M127">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/2013/1/34/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M162">View MathML</a> independent of ε such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M163">View MathML</a>

So, the degree is well defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M160">View MathML</a>. By Lemma 3.3, we can easily see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M165">View MathML</a> admits no solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M160">View MathML</a>. Then, by the homotopy invariance of the Leray-Schauder degree, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M167">View MathML</a>

The proof is completed. □

Now, we show the proof of the main result of this paper.

Proof of Theorem 1

Using Corollaries 1 and 2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M168">View MathML</a>

where R and r are positive constants and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M169">View MathML</a>. Problem (2.2)-(2.4) admits a nonnegative nontrivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M50">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M171">View MathML</a>. Combining with the regularity results [9] and a similar argument as in [10], we can prove that the limit function of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/34/mathml/M50">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 by the 985 project of Harbin Institute of Technology.

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