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The estimates on the energy functional of an elliptic system with Neumann boundary conditions

Jing Zeng

Author Affiliations

School of Mathematics and Computer Sciences, Fujian Normal University, Fuzhou, 350007, P.R. China

Boundary Value Problems 2013, 2013:194  doi:10.1186/1687-2770-2013-194


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/194


Received:19 June 2013
Accepted:13 August 2013
Published:28 August 2013

© 2013 Zeng; 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 consider an elliptic system of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M2">View MathML</a> in Ω with Neumann boundary conditions, where Ω is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M3">View MathML</a> domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M4">View MathML</a>, f and g are nonlinearities having superlinear and subcritical growth at infinity. We prove the existence of nonconstant positive solutions of the system, and estimate the energy functional on a configuration space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M5">View MathML</a> by a different technique, which is an important step in the proof of the solution’s concentrative property. We conclude that the least energy solutions of the system concentrate at the point of boundary, which maximizes the mean curvature of Ω.

Keywords:
elliptic system; estimates; energy functional

1 Introduction

We are concerned with the following singularly perturbed system with Neumann conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M6">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M7">View MathML</a> is a small parameter, Ω is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M3">View MathML</a> bounded domain in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M4">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M10">View MathML</a>), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M11">View MathML</a>. f and g are nonlinearities having superlinear and subcritical growth at infinity.

Problem (1.1) arises in many applied models concerning biological pattern formations. For example, the steady states in the Keller-Segel model, the Gierer-Meinhardt model, see [1,2] for more details. Problem (1.1) has been studied extensively for last twenty years. The motivation for the study of such a problem goes back to the pioneering work of [2,3] concerning the scalar case (single equation),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M12">View MathML</a>

(1.2)

They proved a priori estimates, existence of least energy solutions and the concentrative properties of the solution. Furthermore, in [4,5], Ni and Takagi proved the existence of a nontrivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M13">View MathML</a> to problem (1.2) for ε small enough. They showed that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M13">View MathML</a> attains its maximum value at a point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M15">View MathML</a>, and the subsequences of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M16">View MathML</a> converge to P, which is the maximum point of mean curvature on Ω.

The subject was studied by many authors for both Neumann and Dirichlet boundary conditions. There are many well-known results about (1.2). Del Pino and Felmer in [6] introduced shorter and more elementary arguments with respect to those in [4,7]. Wang in [8] obtained multiple solutions of (1.2) by using Ljusternik-Schnirelman method. In [9], Grossi et al., obtained a solution of (1.2) with k maxima points, k is a given positive integer. We refer the reader to [10-14] for further references.

As far as we know, Avial and Yang [15] were the first to approach the singularly perturbed system (1.1) with Neumann boundary conditions; they considered (1.1) with special nonlinearities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M18">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M19">View MathML</a>). By means of a dual variational formulation, they proved that there exist nontrivial positive solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M21">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M22">View MathML</a>, which have global maximum point at different points.

A more direct approach was proposed in [16-18]. In these papers, the authors extend the idea, which is introduced by Del Pino and Felmer in [6], to system (1.1). In [18], Pistoia and Ramos proved the least energy solutions of system (1.1) concentrate at a point of the boundary, which maximizes the mean curvature of the boundary of Ω. Pistoia and Ramos [19] consider system (1.1) with Dirichlet boundary condition, they proved the existence of the least energy solutions. The solutions are concentrated, as ε goes to zero, at a point of Ω, which is maximized in distance to the boundary of Ω.

Let us recall the idea mentioned in [3-5], their proof based on the well-known result of Gidas et al.[20], that is, the uniqueness solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M23">View MathML</a>

(1.3)

here w is radially symmetric and is strictly decreasing, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M24">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M25">View MathML</a>. But the uniqueness result for the following system corresponding to (1.1) is not known.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M26">View MathML</a>

Besides, it is known that the underlying minimax theorem associated to ground-state level of (1.1) is an infinite-dimensional linking, this is in contrast with (1.2). We refer the reader to [18,21] for more details on this.

In this paper, we prove the existence of nonconstant positive solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M27">View MathML</a> of system (1.1), and estimate the energy functional of (1.1) on the configuration space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M5">View MathML</a> (defined in Section 2) by a different technique, which is compared with [18]. This estimation is an important step in the proof of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M29">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M30">View MathML</a> denotes the mean curvature of Ω at the boundary point P. We conclude the least energy solutions of system (1.1), concentrated at the point of boundary, which maximizes the mean curvature of the boundary of Ω.

2 Statement of main results

The assumption to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M31">View MathML</a> is a typical superlinear subcritical one, as in [18], we assume that the following holds.

(S1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M32">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M33">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M35">View MathML</a>. There exist two real numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M37">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M38">View MathML</a>

(S2) For some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M41">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M42">View MathML</a>

(S3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M44">View MathML</a>, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M41">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M48">View MathML</a>.

Remark 2.1 Examples of nonlinearities satisfying (S1)-(S3) are

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M49">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M51">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M52">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M37">View MathML</a>.

We should point out that (S1)-(S3) are the natural extension of the assumptions for the scalar case (single equation). Let us recall the assumptions on single equation such as (1.2).

Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M54">View MathML</a> is continuous and satisfies the following structure assumptions.

(f1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M55">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M56">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M57">View MathML</a> near <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M58">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M59">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M60">View MathML</a>, for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M61">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M62">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M63">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M64">View MathML</a>.

(f2) There exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M65">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M66">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M67">View MathML</a>, in which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M68">View MathML</a>.

(f3) The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M69">View MathML</a> is strictly increasing.

Remark 2.2 Assumption (S1) is the ‘system edition’ of (f1). (f2) is the famous Ambrosetti-Rabinowitz superlinear condition [22], which has appeared in most of studies for superlinear problems. In fact, it implies that the super-quadratic condition on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M70">View MathML</a>. It has been used in a crucial way not only in establishing the mountain-pass geometry of the functional, but also in obtaining bounds of (PS) sequences. Assumption (S2) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M71">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M72">View MathML</a>, which play a important roll in the proof of the existence of system’s solutions. So it is the ‘system edition’ of (f2).

Without loss of generally, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M11">View MathML</a>. By the following rescaling:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M74">View MathML</a>

equation (1.1) becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M75">View MathML</a>

(2.1)

To simplify the notations, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M76">View MathML</a>. Associated with (2.1) is the energy functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M77">View MathML</a>

(2.2)

(2.2) is a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M3">View MathML</a> functional defined over the Hilbert space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M79">View MathML</a>.

We define the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M80">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M81">View MathML</a>.

It can be observed that the following orthogonal splitting holds: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M82">View MathML</a>, here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M83">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M84">View MathML</a>. We set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M85">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M86">View MathML</a>.

Theorem 2.3Assume (S1)-(S3), then there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M87">View MathML</a>, such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M88">View MathML</a>, system (1.1) has nonconstant positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M89">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M90">View MathML</a>, the estimation of the energy functional on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M5">View MathML</a>is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M92">View MathML</a>

(2.3)

Remark 2.4 We point out that, in contrast with Theorem 2.3, only constant positive solutions are expected to exist for large values of ε[3,15].

Remark 2.5 The estimation (2.3) is an important step in the proof of

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M93">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M30">View MathML</a> denotes the mean curvature of Ω at the boundary point P. So we can conclude the least energy solutions of system (1.1) concentrate at a point of the boundary, which maximizes the mean curvature of the boundary of Ω.

3 Proof of Theorem 2.3

To prove Theorem 2.3, we need the following lemma. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M96">View MathML</a> be the solutions of (1.1), in order to simplify the notations, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M97">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M98">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M99">View MathML</a>

(3.1)

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M101">View MathML</a>.

Lemma 3.1Supposing the assumptions in Theorem 2.3 hold, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M102">View MathML</a>admits the following properties:

(i)1<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M104">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M105">View MathML</a>,

(ii)1<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M106">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M107">View MathML</a>,

(iii)1<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M108">View MathML</a>.

Proof Proof of (i)1. By the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M110">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M111">View MathML</a>, where u and v are the solutions of (1.1), that follows <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M103">View MathML</a>.

Again by (3.1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M113">View MathML</a>

by (S2), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M114">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M115">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M104">View MathML</a>.

Next, we want to show <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M105">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M118">View MathML</a>

(3.2)

From (S2), we can deduce there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M39">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M120">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M121">View MathML</a>

The left side in last inequality is equal to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M122">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M123">View MathML</a>. By the same way, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M124">View MathML</a>. Then (3.2) can change to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M125">View MathML</a>

So, we get the properties (i)1.

To prove (ii)1 is equal to show

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M126">View MathML</a>

We only need to prove for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M127">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M128">View MathML</a>

(3.3)

By (S1), there exist C such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M129">View MathML</a>

As the same,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M130">View MathML</a>

We prove (3.3) by contradiction. Assume that there exists a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M131">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M132">View MathML</a>

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M133">View MathML</a>

So, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M134">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M135">View MathML</a>

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M136">View MathML</a> in the last equality, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M137">View MathML</a>

For p is a number between 2 and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M138">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M139">View MathML</a>, we deduce <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M140">View MathML</a>. It contradicts with the original assumption of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M141">View MathML</a>.

Now, we turn to the prove of (iii)1. By (3.1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M142">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M143">View MathML</a>

take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M144">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M145">View MathML</a>

(3.4)

By (S2), for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M39">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M147">View MathML</a>

in the same way,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M148">View MathML</a>

Thus, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M108">View MathML</a>. □

Proposition 3.2 (Theorem 1.1 in [18])

Under assumptions (H), there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M87">View MathML</a>such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M88">View MathML</a>, problem (1.1) has nonconstant positive solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M152">View MathML</a>. Moreover, both functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M13">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M96">View MathML</a>attain their maximum value at some unique and common point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M155">View MathML</a>. (The assumption (H) is composed of (S1), (S3) and the following (3.5).)

Remark 3.3 We will compare our assumptions (S1)-(S3) with the conditions (H) of Proposition 3.2 in the following proof of Theorem 2.3.

Proof of Theorem 2.3 The existence of solutions of (1.1) can follow the steps of Theorem 1.1 in [18]. They use some ideas introduced by Del Pino and Felmer [6], and differ from the method of Ni and Takagi. It needs to be pointed out that (S2) implies the following conditions:

For some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M39">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M41">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M159">View MathML</a>

(3.5)

The assumption (H) in Proposition 3.2 is composed of (S1), (S3) and (3.5). By Proposition 3.2, the existence of solutions can be proved under (H). So, we can get the existence of solutions of (1.1) under (S1)-(S3).

The rest of the paper is devoted to the proof of (2.3). By the definition of space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M5">View MathML</a>, we only need to prove for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M134">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M163">View MathML</a> the following holds,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M164">View MathML</a>

(3.6)

Obviously <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M165">View MathML</a>. By (3.1), (3.6) is equal to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M166">View MathML</a>

(3.7)

We prove (3.7) by contradiction, suppose that the maximum point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M102">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M168">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M169">View MathML</a>.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M170">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M171">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M172">View MathML</a>

(3.8)

Thus, by (3.8), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M173">View MathML</a>

(3.9)

We claim that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M174">View MathML</a>, defined in (3.9), has the following properties:

(i)2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M175">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M176">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M177">View MathML</a>;

(ii)2 For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M67">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M179">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M180">View MathML</a>;

(iii)2<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M181">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M107">View MathML</a>.

First, we prove (i)2. In fact, by (3.1) and (3.9),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M183">View MathML</a>

Following from assumption (S2), we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M175">View MathML</a>.

Again by (3.9) and Lemma 3.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M185">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M105">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M176">View MathML</a>. Next, we want to compute <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M188">View MathML</a>. From (3.9),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M189">View MathML</a>

(3.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M190">View MathML</a>

(3.11)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M191">View MathML</a>

Combined with (3.4), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M192">View MathML</a>. By Lemma 3.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M108">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M177">View MathML</a>.

Now, we turn to the proof of (ii)2. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M179">View MathML</a>, by (3.11),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M196">View MathML</a>

Then by (3.10),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M197">View MathML</a>

(3.12)

By (S3), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M198">View MathML</a>

similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M199">View MathML</a>

then by (3.12), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M180">View MathML</a>. We proved the property (ii)2.

The proof of (iii)2 is similar to (ii)1. Then we complete the proof of the claim.

Suppose that the maximum point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M102">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M168">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M169">View MathML</a>, then either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M204">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M205">View MathML</a>. If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M206">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M207">View MathML</a>, by (3.1), (3.9) and property (i)2 of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M208">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M209">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M175">View MathML</a>. By the property (iii)2 of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M208">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M212">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M213">View MathML</a>

(3.13)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M205">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M207">View MathML</a>. By Lemma 3.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M216">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M217">View MathML</a>, so there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M218">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M219">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M220">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M221">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M177">View MathML</a>, so there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M223">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M224">View MathML</a>

(3.14)

In fact, (3.13) and (3.14) is a contradiction to the nature (ii)2 of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M174">View MathML</a>, that is, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M67">View MathML</a>, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M179">View MathML</a>, the value of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/194/mathml/M174">View MathML</a> must be smaller than 0.

Having reached a contradiction, this completes the proof of Theorem 2.3. □

Competing interests

There are no financial competing interests in this manuscript. There are no non-financial competing interests (political, personal, religious, ideological, academic, intellectual, commercial or any other) to declare in relation to this manuscript.

Authors’ contributions

There is only one author in the manuscript. JZ designed the study. She carried out the studies, and drafted the manuscript. The author read and approved the final manuscript.

Acknowledgements

The author is supported by the project of ‘Youth Innovation,’ funded by the Department of Science and Technology of Fujian province (2011J05003), and supported by the Projects A of the Educational Department of Fujian Province (JA11053).

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