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Existence of nonnegative nontrivial periodic solutions to a doubly degenerate parabolic equation with variable exponent

Zhongqing Li* and Wenjie Gao

Author Affiliations

College of Mathematics, Jilin University, Changchun, 130012, PR China

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Boundary Value Problems 2014, 2014:77  doi:10.1186/1687-2770-2014-77

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/77


Received:19 October 2013
Accepted:20 March 2014
Published:2 April 2014

© 2014 Li and Gao; 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 credited.

Abstract

The authors investigate a degenerate parabolic equation with delay and nonlocal term, which describes slow diffusive processes in physics or biology. The existence of a nonnegative nontrivial periodic solution is obtained through the use of the Leray-Schauder degree method.

MSC: 35D05, 35K55.

Keywords:
degenerate parabolic equation; periodic solution; variable exponent; topological degree; De Giorgi iteration

1 Introduction

In this paper, we are interested in the following evolutional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M1">View MathML</a>-Laplacian equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M2">View MathML</a>

(1.1)

Here, Ω is a bounded simply connected domain with smooth boundary Ω in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M6">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M7">View MathML</a>. We assume <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M9">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M12">View MathML</a> and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M14">View MathML</a> can be extended as T-periodic functions to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M15">View MathML</a>. Furthermore, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M16">View MathML</a> for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M17">View MathML</a>.

Equation (1.1) is a doubly degenerate parabolic equation with delay and nonlocal term, which models diffusive periodic phenomena in physics and mathematical biology. In biology, it arises from population model, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M18">View MathML</a> denotes the density of population at time t located at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M19">View MathML</a>, a is the natural growth rate of the population, the nonlocal term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M20">View MathML</a> evaluates a weighted fraction of individual, and the delayed density u at time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M21">View MathML</a> appearing in the nonlocal term represents the time needed to an individual to become adult. In physics, problem (1.1) is proposed based on some evolution phenomena in electrorheological fluids [1]. It describes the ability of a conductor to undergo significant changes when an electric field is imposed on. This model has been employed for some technological applications, such as medical rehabilitation equipment and shock wave absorber.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M1">View MathML</a> is a constant and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M24">View MathML</a>, the model describes the slow diffusion process in physics, which has been extensively investigated; see [2-7]. For example, in [5], the authors studied the following doubly degenerate parabolic equation with logistic periodic sources:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M25">View MathML</a>

They proved the existence of a nontrivial nonnegative periodic solution via monotonicity method. Using a Moser iterative method (see [8-11]), they also obtained some a priori bounds and asymptotic behaviors for the solutions.

Recently, the variable exponent Sobolev space and its applications have attracted considerable interest; see [1,12-14] and the references therein. When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M8">View MathML</a>, the doubly degenerate parabolic equation (1.1) is a more realistic model which describes the rather slow diffusion process. In our models, the principal term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M28">View MathML</a>, in place of the usual term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M29">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M30">View MathML</a>, represents nonhomogeneous diffusion that depends on the position <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M31">View MathML</a> and thus gives a better description of nonhomogeneous character of the process.

There are many differences between Sobolev spaces with constant exponent and those with variable exponent; many powerful tools applicable in constant exponent spaces are not available for variable exponent spaces. For instance, the variable exponent spaces are no longer translation invariant and Young’s inequality <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M32">View MathML</a> holds if and only if p is constant (see monograph [12]). As we all know, the frequently used Hölder’s inequality, Poincaré’s inequality, etc., will be presented in new forms for variable exponent spaces.

The presence of the nonlocal term and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M1">View MathML</a>-Laplacian term makes the sup-solution and sub-solution method (as in [5]) in vain. In our paper, we adopt the topological degree method (as in [8-10]) to show the existence of nontrivial periodic solutions to problem (1.1). However, the method employed in the variable exponent case [13] or in the constant exponent case [8-11] cannot be directly used to derive the uniform upper bound for solutions, which is a crucial step in applying the topological degree method. We apply a modified De Giorgi iteration to establish the crucial uniform bound. We believe that the modified De Giorgi iteration used in this paper can be employed to other types equations with nonstandard growth conditions.

We now discuss the main plan of the paper. In Section 2, we review some preliminaries concerning the variable exponent Sobolev spaces and introduce a family of regularized problems for problem (1.1). We regularize the degenerate part through replacing the term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M34">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M35">View MathML</a>

In Section 3, in order to apply the topological degree method, we combine these regularized problems with a relatively simpler equation and derive some a priori estimates. By virtue of the De Giorgi iteration technique, we deduce an a priori <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M36">View MathML</a> bound for solutions to the regularized problems in Proposition 3.2; and the uniform lower bound estimate is obtained in Proposition 3.5. In Section 4, we establish the existence of nonnegative nontrivial solution of (1.1) through the limit process as ϵ and η tend to zero. Finally, in the Appendix, we give a proof of the iteration lemma (Lemma 3.1) for the sake of readability.

2 Preliminaries and the regularized problems of (1.1)

First of all, for the reader’s convenience, we recall some preliminary results concerning the variable exponent Sobolev spaces. One may find these standard results in monographs [1,12].

Let p be a continuous function defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M38">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M39">View MathML</a>.

1. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M40">View MathML</a>space: We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M41">View MathML</a>

equipped with the following Luxemburg norm:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M42">View MathML</a>

The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M43">View MathML</a> is a separable, uniformly convex Banach space.

2. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M44">View MathML</a>space: We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M45">View MathML</a>

endowed with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M46">View MathML</a>. We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M47">View MathML</a> the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M48">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M44">View MathML</a>. In fact, the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M50">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M51">View MathML</a> are equivalent norms in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M47">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M53">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M47">View MathML</a> are separable and reflexive Banach spaces with the above norms.

3. Frequently used relationships in the estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M55">View MathML</a>

4. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M1">View MathML</a>-Hölder’s inequality:

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M57">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M58">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M59">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M60">View MathML</a>

5. Embedding relationships:

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M61">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M62">View MathML</a> are in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M63">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M64">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M39">View MathML</a>, then there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M66">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M67">View MathML</a>

i.e. the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M68">View MathML</a> is continuous.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M69">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M70">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M71">View MathML</a>, then the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M72">View MathML</a> is continuous and compact. Here

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M73">View MathML</a>

6. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M1">View MathML</a>-Poincaré’s inequality:

There exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M75">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M76">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M77">View MathML</a>.

We next define the weak solutions to problem (1.1).

Definition 2.1u is said to be a weak periodic solution to (1.1) provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M78">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M79">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M80">View MathML</a> and u satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M81">View MathML</a>

(2.1)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M82">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M83">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M85">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M86">View MathML</a>.

As in [7], we introduce the following regularized problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M87">View MathML</a>

(2.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M90">View MathML</a> are given constants.

Definition 2.2 We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91">View MathML</a> is a weak periodic solution of (2.2), if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M92">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M94">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91">View MathML</a> solves

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M96">View MathML</a>

(2.3)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M82">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M83">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M85">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M86">View MathML</a>.

Remark 2.3 For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M9">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M103">View MathML</a> is dense in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M104">View MathML</a>, thus in the sense of the definition of weak solution above, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91">View MathML</a> can be chosen as test function.

We investigate problem (2.2) extensively before studying the limit process as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M106">View MathML</a>. Define a map <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M107">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M108">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91">View MathML</a> is a weak periodic solution of the problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M110">View MathML</a>

(2.4)

Given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M111">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M112">View MathML</a> be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M113">View MathML</a>

Therefore, if a nonnegative function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M114">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M115">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91">View MathML</a> is a weak solution of (2.2).

Define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M117">View MathML</a>

Then, according to [3] or the classical regularity results from [4], one obtains the following lemma.

Lemma 2.4Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M88">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M119">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M120">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M121">View MathML</a>is a continuous compact operator from<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M122">View MathML</a>to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M123">View MathML</a>. Furthermore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M124">View MathML</a>.

3 A priori estimates to the regularized problem

First of all, the following modified De Giorgi iteration lemma will be useful (we give a proof in the Appendix).

Lemma 3.1 (Iteration lemma)

Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M125">View MathML</a>is a nonnegative and nonincreasing function on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M126">View MathML</a>, it satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M127">View MathML</a>

(3.1)

for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M128">View MathML</a>, and for some constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M130">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M131">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M132">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M133">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M134">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M135">View MathML</a>.

Next, we prove a crucial a priori <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M36">View MathML</a> bound for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M137">View MathML</a> via a De Giorgi iteration technique as in [15].

Proposition 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M138">View MathML</a>and assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91">View MathML</a>is a nonnegativeT-periodic continuous function such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M140">View MathML</a>

(3.2)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M141">View MathML</a>

(3.3)

Then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M142">View MathML</a>, such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M143">View MathML</a>, whereRis independent ofϵandη.

ProofStep 1. Multiplying (3.2) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M144">View MathML</a>, with any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M145">View MathML</a>. Integrating over Ω and noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M146">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M147">View MathML</a>

(3.4)

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M148">View MathML</a>, we deal with the second term on the left-hand side of (3.4) as follows.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M149">View MathML</a>

(3.5)

Combining (3.4) and (3.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M150">View MathML</a>

(3.6)

We estimate the right-hand side of (3.6) by Hölder’s inequality, the embedding theorem and Young’s inequality with ϵ to deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M151">View MathML</a>

(3.7)

Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M152">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M153">View MathML</a> appropriately, we have from (3.6) and (3.7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M154">View MathML</a>

(3.8)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M145">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M156">View MathML</a> depends on q, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M157">View MathML</a>, m, and Ω.

Integrating (3.6) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M158">View MathML</a> and using the T-periodicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M160">View MathML</a>

(3.9)

Similarly to (3.7), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M161">View MathML</a>

(3.10)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M162">View MathML</a> depends on q, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M157">View MathML</a>, m, T and Ω. By Poincaré’s inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M164">View MathML</a>

(3.11)

Recall our assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M165">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M8">View MathML</a>, and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M167">View MathML</a>. Consequently, considering (3.11), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M168">View MathML</a>

(3.12)

which implies that there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M169">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M170">View MathML</a>

(3.13)

From (3.8) and (3.13), we conclude

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M171">View MathML</a>

(3.14)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M172">View MathML</a>. In view of the T-periodicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M137">View MathML</a>, (3.14) shows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M174">View MathML</a>

We finally arrive at

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M175">View MathML</a>

(3.15)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M145">View MathML</a>, where C depends on q, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M157">View MathML</a>, m, T and Ω.

Step 2. Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M178">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M179">View MathML</a> is the Lebesgue measure of the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M180">View MathML</a>. Multiplying (3.2) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M181">View MathML</a> on both sides, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M182">View MathML</a> represents the characteristic function of the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M183">View MathML</a>, and integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M184">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M185">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M186">View MathML</a>. We assume that the absolutely continuous function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M187">View MathML</a> attains its maximum at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M188">View MathML</a>. Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M189">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M190">View MathML</a> and θ small enough so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M191">View MathML</a>. (In fact, this is always possible because of the periodicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91">View MathML</a>; for example, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M193">View MathML</a>, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M194">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M195">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M191">View MathML</a>.) Then we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M197">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M198">View MathML</a>

(3.16)

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M199">View MathML</a> yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M200">View MathML</a>

(3.17)

After a direct computation, we obtain an estimate for the left-hand side of (3.17) as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M201">View MathML</a>

(3.18)

Substituting (3.18) into (3.17), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M202">View MathML</a>

(3.19)

We now deal with (3.19). On one hand, by the embedding theorem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M203">View MathML</a>

(3.20)

where S is the Sobolev embedding constant, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M204">View MathML</a>

On the other hand, from (3.15), where we may fix a special q, using Hölder’s inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M205">View MathML</a>

(3.21)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M206">View MathML</a>. Then (3.19), (3.20), and (3.21) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M207">View MathML</a>

(3.22)

Utilizing Young’s inequality with ϵ, we obtain from (3.22)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M208">View MathML</a>

Upon choosing ϵ appropriately, one obtains

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M209">View MathML</a>

(3.23)

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M210">View MathML</a>, it is easy to see

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M211">View MathML</a>

(3.24)

The relationships (3.23) and (3.24) above imply that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M212">View MathML</a>

(3.25)

Noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M213">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M214">View MathML</a>, by the iteration Lemma 3.1, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M215">View MathML</a> and thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M216">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M217">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M218">View MathML</a>

 □

Theorem 3.3Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M16">View MathML</a>, for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M17">View MathML</a>. Then there exists a positive constantRsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M221">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M222">View MathML</a>.

Proof From Proposition 3.2, we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M223">View MathML</a>, it implies that there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M142">View MathML</a> independent of ϵ and η, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M225">View MathML</a>, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M226">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M227">View MathML</a>. Hence the topological degree <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M228">View MathML</a> is well defined in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M229">View MathML</a>. Thanks to the homotopy invariance property of the Leray-Schauder degree, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M230">View MathML</a>

(3.26)

Using the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M231">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M232">View MathML</a>

(3.27)

From (3.26) and (3.27), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M233">View MathML</a>. □

Using the standard method, similar to that in [3] or [13], one can prove the following.

Proposition 3.4Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M14">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M236">View MathML</a>solves<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M237">View MathML</a>, for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M90">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M227">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M240">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M17">View MathML</a>. Moreover, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M242">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M243">View MathML</a>in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M184">View MathML</a>.

In what follows, we prove a lower bound for the regularized problem.

Proposition 3.5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M245">View MathML</a>be the first eigenvalue of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M246">View MathML</a>

and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M247">View MathML</a>be the associated positive eigenfunction such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M248">View MathML</a>. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M249">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M250">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M251">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M242">View MathML</a>satisfies<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M253">View MathML</a>for some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M90">View MathML</a>, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M255">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M256">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M257">View MathML</a>is the embedding constant of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M258">View MathML</a>into<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M40">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M260">View MathML</a>is the Lebesgue measure of the domain Ω.

Proof We argue by contradiction. If not, then for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M90">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M262">View MathML</a>, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M242">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M264">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M265">View MathML</a>. For clarity, we divide the proof into four steps.

Step 1. Note that, by Proposition 3.4, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M243">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M184">View MathML</a>. Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M268">View MathML</a> and multiplying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M269">View MathML</a>

(3.28)

by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M270">View MathML</a>, integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M184">View MathML</a> and using the T-periodicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M272">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M273">View MathML</a>

(3.29)

Step 2. Using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M274">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M275">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M276">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M88">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M278">View MathML</a>. Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M279">View MathML</a>

(3.30)

Thanks to the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M1">View MathML</a>-Hölder’s inequality in variable exponent space, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M281">View MathML</a>

(3.31)

Noting that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M282">View MathML</a> and using Hölder’s inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M283">View MathML</a>

(3.32)

Integrating (3.31) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M284">View MathML</a> and noting (3.32), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M285">View MathML</a>

(3.33)

Step 3. Multiplying (3.28) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M286">View MathML</a>, integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M184">View MathML</a>, noticing the T-periodicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M289">View MathML</a>, we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M290">View MathML</a>

Substituting this inequality into (3.33), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M291">View MathML</a>

(3.34)

Substituting (3.34) into (3.30) and noticing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M292">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M293">View MathML</a>

(3.35)

Considering that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M251">View MathML</a>, from (3.29) and (3.35), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M295">View MathML</a>

(3.36)

Step 4. We claim

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M296">View MathML</a>

(3.37)

from which we will derive a contradiction. First, to show (3.37), let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M297">View MathML</a> in (3.36). Using the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M298">View MathML</a> and noting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M248">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M300">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M301">View MathML</a>

Now the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M302">View MathML</a> and (3.37) yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M303">View MathML</a>

(3.38)

which is clearly a contradiction to the assumption that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M262">View MathML</a>. This completes the proof. □

Theorem 3.6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M302">View MathML</a>be as given in Proposition 3.5. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M306">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M307">View MathML</a>.

Proof In view of Proposition 3.5, for any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M308">View MathML</a>, we have proved that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M309">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M310">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M90">View MathML</a>. So the Leray-Schauder topological degree <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M312">View MathML</a> is well defined for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M90">View MathML</a>. Thanks to the homotopy invariance of the topological degree, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M314">View MathML</a>

(3.39)

Also, from Proposition 3.5, we infer that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M315">View MathML</a> admits no nontrivial solution in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M316','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M316">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M317">View MathML</a> is not a solution of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M315">View MathML</a>. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M319">View MathML</a>. Together with (3.39), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M320">View MathML</a>. □

4 Existence of nontrivial nonnegative solution to (1.1)

Theorem 4.1Assume<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M321">View MathML</a>for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M17">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M323">View MathML</a>. Then problem (1.1) has a nontrivial nonnegative periodic solution.

Proof We consider the regularized problem (2.2). By Theorem 3.3 and Theorem 3.6, we conclude that there exist R and r, independent of ϵ and η, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M324">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M325">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M88">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M251">View MathML</a>. Using the solvability of the Leray-Schauder degree, we conclude that the regularized problem (2.2) admits a nontrivial nonnegative solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M329">View MathML</a>.

We prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M330">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M93">View MathML</a> and that a solution to problem (1.1) is obtained as a limit of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M333">View MathML</a>. We proceed in several steps.

Step 1. In view of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M321">View MathML</a>, choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M335">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M336">View MathML</a>

(4.1)

Multiplying (4.1) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M337">View MathML</a>, integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M184">View MathML</a> and noting the T-periodicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91">View MathML</a> and the boundedness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M341">View MathML</a>

(4.2)

where M is a positive constant independent of ϵ and η. Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M342">View MathML</a>

(4.3)

So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M330">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M344">View MathML</a> is uniformly bounded in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M345">View MathML</a>. Thus, up to subsequence if necessary, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M346">View MathML</a>. In what follows, our main goal is to prove that u is a weak solution of problem (1.1).

Step 2. The following relation is obvious:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M347">View MathML</a>

(4.4)

From (4.2) and (4.4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M348">View MathML</a>

(4.5)

Owing to the embedding results in the variable exponent space, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M349">View MathML</a>

(4.6)

Integrating (4.6) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M284">View MathML</a> and using Hölder’s inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M351">View MathML</a>

(4.7)

From (4.5) and (4.7), there exists a positive constant C independent of ϵ and η, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M352','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M352">View MathML</a>

(4.8)

In the following, we prove

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M353">View MathML</a>

(4.9)

First, denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M354">View MathML</a>

A straightforward computation shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M355">View MathML</a>

(4.10)

By the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M1">View MathML</a>-Hölder’s inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M357">View MathML</a>

(4.11)

Integrating (4.11) over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M284">View MathML</a>, using the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M1">View MathML</a>-Hölder’s inequality again, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M360">View MathML</a>

(4.12)

Substituting (4.5), (4.8), and (4.12) into (4.10), we derive (4.9). Therefore, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M361">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M362">View MathML</a>

(4.13)

weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M363">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M333">View MathML</a>.

Step 3. Using a method analogous to [7], we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M365">View MathML</a>, where C is independent of ϵ and η. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M344">View MathML</a> is uniformly bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M367">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M368','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M368">View MathML</a>, by compactness theorem (Corollary 4 in [16]), it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M369">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M370">View MathML</a>. Thus, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M371','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M371">View MathML</a>

(4.14)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M82">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M83">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M19">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M375','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M375">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M86">View MathML</a> (and hence, by density, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M377">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M378">View MathML</a> and T-periodicity). The continuity of u follows from similar Hölder estimates in [17].

Step 4. It remains to verify for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M82">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M380">View MathML</a>

(4.15)

We consider matrix function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M381">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M382">View MathML</a> is a positive definite matrix. Choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M383">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M384">View MathML</a>, by mean value theorem, there exists a matrix Y such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M385','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M385">View MathML</a>

(4.16)

which gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M386">View MathML</a>

(4.17)

Multiplying the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M387','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M387">View MathML</a>

by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M388">View MathML</a>, integrating over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M184">View MathML</a> and using T-periodicity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M91">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M391','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M391">View MathML</a>

(4.18)

Thus, (4.17) and (4.18) imply

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M392">View MathML</a>

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M106">View MathML</a>, by (4.13), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M394">View MathML</a>

(4.19)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M395">View MathML</a> in (4.14) and, by the T-periodicity of u, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M396">View MathML</a>

(4.20)

Combining (4.19) with (4.20), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M397','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M397">View MathML</a>

(4.21)

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M398">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M399">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M82">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M401','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M401">View MathML</a>

(4.22)

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M402">View MathML</a> in (4.22) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M403">View MathML</a>

(4.23)

On the other hand, if we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M404','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M404">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M399','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M399">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M82">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M402','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M402">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M408">View MathML</a>

(4.24)

From (4.23) and (4.24) we have (4.15). This completes the proof of Theorem 4.1. □

Appendix

In this appendix, we prove Lemma 3.1 for the reader’s convenience.

Proof of Lemma 3.1 Define the following sequence:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M409">View MathML</a>

where d is to be determined later. Then (3.1) implies the recursive relationship

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M410">View MathML</a>

(5.1)

By induction, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M411">View MathML</a>

(5.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M412">View MathML</a> is to be chosen. In fact, if (5.2) is right, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M413">View MathML</a>

We choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M414','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M414">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M415','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M415">View MathML</a>. Consequently, these choices guarantee <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M416','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M416">View MathML</a>. From (5.2) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/77/mathml/M125">View MathML</a> is nonnegative and nonincreasing, we have deduced the result. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the manuscript and approved the final version.

Acknowledgements

The authors would like to thank the anonymous referees for their valuable comments on and suggestions regarding the original manuscript. This work was supported by National Science Foundation of China (11271154), by Key Lab of Symbolic Computation and Knowledge Engineering of Ministry of Education and by the 985 program of Jilin University.

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