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Time-periodic solutions for a driven sixth-order Cahn-Hilliard type equation

Changchun Liu*, Aibo Liu and Hui Tang

Author Affiliations

Department of Mathematics, Jilin University, Changchun, 130012, China

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Boundary Value Problems 2013, 2013:73  doi:10.1186/1687-2770-2013-73


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/73


Received:6 November 2012
Accepted:14 March 2013
Published:4 April 2013

© 2013 Liu et al.; licensee Springer

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

Abstract

We study a driven sixth-order Cahn-Hilliard type equation which arises naturally as a continuum model for the formation of quantum dots and their faceting. Based on the Leray-Schauder fixed point theorem, we prove the existence of time-periodic solutions.

MSC: 35B10, 35K55, 35K65.

Keywords:
sixth-order Cahn-Hilliard equation; time-periodic solution; existence; Campanato space

1 Introduction

In this paper, we are concerned with the following problem for the sixth-order Cahn-Hilliard type equation:

(1.1)

(1.2)

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M8">View MathML</a> are Hölder continuous functions defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M9">View MathML</a> with period T, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M10">View MathML</a> belongs to the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M11">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M12">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M13">View MathML</a>. Furthermore, we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M17">View MathML</a>, where γ, ν, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M19">View MathML</a>, N, L and Λ are positive constants.

Equation (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M20">View MathML</a> arises naturally as a continuum model for the formation of quantum dots and their faceting; see [1]. It can also be used to describe competition and exclusion of biological population [2]. If we consider that the perturbation function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M10">View MathML</a> (for example, source) has the influence, then we obtain equation (1.1).

Korzec et al.[3] studied equation (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M20">View MathML</a>. New types of stationary solutions of one-dimensional driven sixth-order Cahn-Hilliard type equation (1.1) are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. Liu et al.[4] proved that equation (1.1) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M20">View MathML</a> possesses a global attractor in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M24">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M25">View MathML</a>) space, which attracts any bounded subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M26">View MathML</a> in the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M24">View MathML</a>-norm.

During the past years, many authors have paid much attention to other sixth-order thin film equations such as the existence, uniqueness and regularity of the solutions [5-7]. However, as far as we know, there are few investigations concerned with the time-periodic solutions of equation (1.1), even though there is some literature for population models and Cahn-Hilliard [8,9]. In fact, it is natural to consider the time-periodic solutions of equation (1.1) when it is used to describe the models of the growth and dispersal in the population which is sensitive to time-periodic factors (for example, seasons). In this paper, we prove the existence of time-periodic solutions of problem (1.1)-(1.3) based on the framework of the Leray-Schauder fixed point theorem which can be found in any standard textbook of PDE (see, for example, [10]). For this purpose, we first introduce an operator ℒ by considering a linear sixth-order equation with a parameter <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M28">View MathML</a>. After verifying the compactness of the operator and some necessary a priori estimates for the solutions, we then obtain a fixed point of the operator in a suitable functional space with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M29">View MathML</a>, which is the desired solution of problem (1.1)-(1.3).

The main difficulties for treating problem (1.1)-(1.3) are caused by the nonlinearity of both the fourth-order term and the convective factors. The main method that we use is based on the Schauder-type a priori estimates, which here are obtained by means of a modified Campanato space. We note that the Campanato spaces have been widely used for the discussion of partial regularity of solutions of parabolic systems of second order and fourth order. So, in the following section we give a detailed description and the associated properties of such a space, and subsequently, in the next section we prove the existence of classical time-periodic solutions of problem (1.1)-(1.3).

2 Hölder norm estimates

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M30">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M31">View MathML</a>. For any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M32">View MathML</a>, we define

Let u be a function defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M34">View MathML</a>, and set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M35">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M36">View MathML</a> denotes the parabolic boundary of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M34">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M38">View MathML</a> denotes the area of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M39">View MathML</a>.

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M40">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M41">View MathML</a>, define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M42">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M43">View MathML</a>. By the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M44">View MathML</a> we mean the subset of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M45">View MathML</a>, each element of which satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M46">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M47">View MathML</a>, its norm is defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M48">View MathML</a>

Now, we give some useful lemmas.

Lemma 2.1[11]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M49">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M50">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M51">View MathML</a>.

Now we consider the following linear periodical problem:

(2.1)

(2.2)

(2.3)

Here we simply assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M55">View MathML</a> is sufficiently smooth. Our main purpose is to find the relation between the Hölder norm of the solution u and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M55">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M31">View MathML</a> be a fixed point and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M58">View MathML</a>

Let u be an arbitrary solution of problem (2.1)-(2.3). We split u on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M59">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M60">View MathML</a> so that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M61">View MathML</a> solves the problem

(2.4)

(2.5)

(2.6)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M65">View MathML</a> solves the problem

(2.7)

(2.8)

(2.9)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M69">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M70">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M71">View MathML</a> are the down-side and up-side points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M72">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M73">View MathML</a> is the boundary of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M74">View MathML</a>.

Some essential estimates on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M61">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M65">View MathML</a> are based on the following lemmas.

Lemma 2.2For the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M65">View MathML</a>of problem (2.7)-(2.9), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M78">View MathML</a>

(2.10)

whereCis a positive constant, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M79">View MathML</a>.

Proof Noticing the condition (2.8) and the boundary value condition (2.9), we use the Poincaré inequality and interpolation method (see Chapter 5 in [12]) and get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M80">View MathML</a>

(2.11)

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M81">View MathML</a>

(2.12)

Multiplying equation (2.7) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M65">View MathML</a>, integrating the result over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M39">View MathML</a> and using the boundary value condition (2.9), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M84">View MathML</a>

(2.13)

Using the Young inequality and (2.12), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M85">View MathML</a>

(2.14)

Combining (2.12), (2.13) and (2.14) yields the estimate (2.10) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M86">View MathML</a>.

Similarly, multiplying (2.7) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M87">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M88">View MathML</a>, we can obtain the estimates (2.10) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M89">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M90">View MathML</a>. □

Lemma 2.3For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M91">View MathML</a>,

(2.15)

(2.16)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M94">View MathML</a>

andCis a constant number. Further, (2.15) and (2.16) still hold if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M61">View MathML</a>is replaced by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M96">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M97">View MathML</a>.

Proof The estimate (2.15) is obvious. In fact, by the Hölder inequality,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M98">View MathML</a>

For (2.16), we only consider the case when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M99">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M100">View MathML</a>. Integrating equation (2.4) over the region <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M101">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M102">View MathML</a>

Integrating the above equation with respect to z over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M103">View MathML</a>, and then integrating the result with respect to y over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M104">View MathML</a>, we have

By virtue of the mean value theorem and the Hölder inequality, we see that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M106">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M107">View MathML</a>

Combining the above result with (2.15), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M108">View MathML</a>

To prove the results on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M96">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M97">View MathML</a>, we only need to differentiate equation (2.4) once or twice with respect to x. And the next procedures are completely similar to the above argument. □

Lemma 2.4

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M111">View MathML</a>

(2.17)

whereCis a constant and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M112">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M113">View MathML</a>

Proof In order to prove (2.17) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M89">View MathML</a>, we first prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M115">View MathML</a>

(2.18)

We discuss it in the following two cases.

(I) We first prove (2.18) in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M116">View MathML</a>. In such a case, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M117">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M118">View MathML</a>. Choose a smooth function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M119">View MathML</a> satisfying the following requirements.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M120">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M121">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M122">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M124">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M125">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M126">View MathML</a>, then the value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M119">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M128">View MathML</a> is changed into 1.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M129">View MathML</a>, then the value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M119">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M131">View MathML</a> is changed into 1.

Multiplying equation (2.4) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M132">View MathML</a> and integrating the result over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M39">View MathML</a>, then using the boundary value condition (2.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M134">View MathML</a>

By the Young inequality and the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M119">View MathML</a>, we have

Similarly, we can estimate other three terms. Combining the above expressions yields

(2.19)

As for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M138">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M139">View MathML</a>

(2.20)

As for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M140">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M141">View MathML</a>

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M142">View MathML</a>

On the other hand,

Combining the above two yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M144">View MathML</a>

(2.21)

Notice that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M145">View MathML</a>

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M146">View MathML</a>

(2.22)

Finally, from (2.20), (2.21) and (2.22), we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M147">View MathML</a>

which combined with (2.19) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M148">View MathML</a>

(2.23)

We can imitate all the above procedures and derive a similar result on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M74">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M150">View MathML</a>

(2.24)

where H is a polynomial with respect to χ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M153">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M154">View MathML</a> and satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M155">View MathML</a>. Using the Sobolev inequality on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M72">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M157">View MathML</a>

Combining the above with (2.24) yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M158">View MathML</a>

Combining the above with (2.23) yields the desired estimate (2.18).

(II) Then we prove (2.18) in the case 0 or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M159">View MathML</a>. Take the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M160">View MathML</a> as an example. Choose another smooth function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M161">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M162">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M163">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M164">View MathML</a> when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M165">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M166">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M167">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M168">View MathML</a>.

With λ stated in the lemma, we multiply (2.4) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M169">View MathML</a> and integrate the result over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M39">View MathML</a>. Then we can derive equalities similar to the above argument in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M61">View MathML</a> is replaced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M172">View MathML</a> and a term

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M173">View MathML</a>

is added. Then following the argument as in Case I, we can complete the proof of (2.18).

Now we multiply (2.4) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M174">View MathML</a> and follow the above argument. Then we derive the same result on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M96">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M176">View MathML</a>

(2.25)

Using the interpolation inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M177">View MathML</a>

Replacing R in (2.23) by 2R, and combining the result with the above inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M178">View MathML</a>

which together with (2.25) yields (2.17) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M89">View MathML</a>.

For (2.17) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M90">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M181">View MathML</a>, we should first multiply (2.4) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M174">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M183">View MathML</a> respectively, and the remaining parts are similar and easier. □

Lemma 2.5For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M184">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M185">View MathML</a>

(2.26)

whereCis a constant number. Further, (2.26) still holds, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M61">View MathML</a>is replaced by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M96">View MathML</a>or<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M97">View MathML</a>.

Proof It suffices to show (2.26) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M189">View MathML</a>, otherwise we only need to set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M190">View MathML</a>. By Lemma 2.3 and Lemma 2.4, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M191">View MathML</a>

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M192">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M193">View MathML</a>

(2.27)

On the other hand, by (2.25),

which combined with (2.27) implies (2.26). The proofs of the results on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M96">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M97">View MathML</a> are similar. □

Lemma 2.6Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M197">View MathML</a>be a nonnegative and nondecreasing function satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M198">View MathML</a>

whereA, B, α, βare positive constants and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M199">View MathML</a>. Then there exists a constantConly depending onA, B, α, βsuch that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M200">View MathML</a>

The proof of this lemma can be found in [13].

Theorem 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M55">View MathML</a>be an appropriately smooth function, and letube the smooth solution of problem (2.1)-(2.3). Then, for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M202">View MathML</a>, there exists a coefficientKdepending only onα, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M203">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M204">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M205">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M206">View MathML</a>

(2.28)

Further, (2.28) still holds ifuis replaced byDuor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M207">View MathML</a>.

Proof For any fixed point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M208">View MathML</a>, consider the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M209">View MathML</a>, which is clearly nondecreasing with respect to ρ. By Lemma 2.5,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M210">View MathML</a>

holds for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M184">View MathML</a>. By Lemma 2.2,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M212">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M213">View MathML</a>

By Lemma 2.6, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M214">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M215">View MathML</a>. Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M216">View MathML</a>

Using Lemma 2.1, we immediately obtain (2.28). The proofs of the results on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M96">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M97">View MathML</a> are similar. □

3 The main result and its proof

In this section, we represent the main result of this paper.

Theorem 3.1Problem (1.1)-(1.3) admits a time-periodic solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M219">View MathML</a>.

To prove the existence of this solution, we employ the Leray-Schauder fixed point theorem which enables us to study the problem by considering the following equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M220">View MathML</a>

(3.1)

subject to the conditions (1.2)-(1.3), where σ is a parameter taking value on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M221">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M222">View MathML</a> is periodic in time t with period T, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M223">View MathML</a>. For any given function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M224">View MathML</a>, from linear classical theory (see [14]), we see that problems (3.1) and (1.2)-(1.3) admit a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M225">View MathML</a>, and hence we can define a mapping ℒ as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M226">View MathML</a>

together with its composition with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M227">View MathML</a>, namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M228">View MathML</a>

Obviously, for any given <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M229">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M230">View MathML</a>. By virtue of the Leray-Schauder fixed point theorem, to prove the existence of solutions of problem (1.1)-(1.3), we only need to show that the mapping ℒ is compact and prove that there exists a constant independent of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M231">View MathML</a> and σ such that, for any u and σ satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M233">View MathML</a>. Moreover, it follows from the above arguments that u is a classical solution. Then we consider the problem in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M234">View MathML</a> in turn. Finally, we know that initial boundary value problem (1.1)-(1.3) admits a classical solution in Q.

Lemma 3.1The mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M235">View MathML</a>is compact.

This result can be directly obtained by a compact embedding theorem, so we omit the details here.

Lemma 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M231">View MathML</a>be a time-periodic solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M237">View MathML</a>

(3.2)

subject to the conditions (1.2)-(1.3), where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M28">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M239">View MathML</a>

(3.3)

whereCis a constant independent of the solutionuandσ.

Proof First, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M240">View MathML</a> be a time-periodic solution of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M241">View MathML</a>

then from the Poincaré inequality we know that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M242">View MathML</a>

(3.4)

Multiplying (3.2) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M240">View MathML</a>, integrating the result over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M34">View MathML</a> and using the condition (1.2), then using the Young inequality and (3.4), we have

(3.5)

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M246">View MathML</a>

(3.6)

Moreover,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M247">View MathML</a>

(3.7)

It follows from (3.5) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M248">View MathML</a>

(3.8)

By (1.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M249">View MathML</a>

Integrating the above inequality over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M34">View MathML</a> and using (3.8) together with the Young inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M251">View MathML</a>

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M252">View MathML</a>

On the other hand, by the Young inequality,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M253">View MathML</a>

Combining the above expressions, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M254">View MathML</a>

(3.9)

Combining the above with (3.6) and (3.7), we see that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M255">View MathML</a>

(3.10)

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M256">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M257">View MathML</a>, λ is a positive constant depending only on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M18">View MathML</a> and N. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M259">View MathML</a>. Integrating <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M260">View MathML</a> over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M261">View MathML</a>, by (3.9) and (3.10), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M262">View MathML</a>

(3.11)

On the other hand, integrating by parts and using (1.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M263">View MathML</a>

Integrating the above equality over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M261">View MathML</a> and noticing the periodicity of F, we have

Integrating <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M266">View MathML</a> over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M261">View MathML</a>, using (3.9) and (3.10), we have

(3.12)

By virtue of (3.11) and (3.12), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M269">View MathML</a>. Noticing the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M270">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M271">View MathML</a>

(3.13)

By (1.2), we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M272">View MathML</a>.

In order to prove the rest of this lemma, we need to give a priori estimate on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M273">View MathML</a>. First, by the Gagliardo-Nirenberg inequality, we can obtain

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M275">View MathML</a> denotes the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M276">View MathML</a> norm on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M277">View MathML</a>. Regulating the exponents and using the Young inequality for every of the above three expressions, we get

Integrating the above inequalities over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M261">View MathML</a> and noticing (3.13), we see that the terms of left hand side in these inequalities can all be estimated by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M280">View MathML</a> and a constant number C. Then by the boundary value condition and (3.10), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M281">View MathML</a>

(3.14)

and also, by the above discussion, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M282">View MathML</a>

(3.15)

Multiplying (3.2) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M283">View MathML</a>, integrating the result over <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M34">View MathML</a>, using (3.14), (3.15) and the Young inequality, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M285">View MathML</a>

(3.16)

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M286">View MathML</a>

(3.17)

By (3.17) and the approach similar to (3.14), we can derive

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M287">View MathML</a>

(3.18)

Now we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M288">View MathML</a>

Obviously,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M289">View MathML</a>

(3.19)

On the other hand, by (3.16), (3.17) and (3.18), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M290">View MathML</a>

(3.20)

By virtue of (3.19) and (3.20), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M291">View MathML</a>. Noticing the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M292">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M293">View MathML</a>

Applying the Poincaré inequality and the Friedrichs inequality [15], we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M294">View MathML</a>.

Finally, we set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M295">View MathML</a>

By an approach similar to the above argument, we can obtain the last result that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M296">View MathML</a>. The proof of this lemma is complete. □

Proof of Theorem 3.1 Now we apply Theorem 2.1 to complete the proof of Theorem 3.1. For the smooth function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M55">View MathML</a> in Theorem 2.1, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M298">View MathML</a>

From the proof of Lemma 3.2, we see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M299">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M300">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M205">View MathML</a> can be all uniformly bounded by a constant number C. Therefore the coefficient K in Theorem 2.1 now only depends on the Hölder exponent α. So, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M231">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M303">View MathML</a>

which combines with the results of Lemma 3.2. We know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M304">View MathML</a>, where C is independent of u and σ. Then, it follows from the results in [16] that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M305">View MathML</a>. Recalling the discourse in the beginning of this section, we conclude from the Leray-Schauder fixed point theorem that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M306">View MathML</a> admits a fixed point u in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/73/mathml/M307">View MathML</a>, which is the desired solution of problem (1.1)-(1.3). The proof of Theorem 3.1 is completed. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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

Acknowledgements

The authors would like to express their deep thanks to the referees for their valuable suggestions, for the revision and improvement of the manuscript. This research was partly supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

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