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

Open Access Research

On symmetric positive homoclinic solutions of semilinear p-Laplacian differential equations

Stepan Tersian

Author Affiliations

Department of Mathematical Analysis, University of Ruse, Ruse, 7017, Bulgaria

Boundary Value Problems 2012, 2012:121  doi:10.1186/1687-2770-2012-121

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/121


Received:31 July 2012
Accepted:8 October 2012
Published:24 October 2012

© 2012 Tersian; licensee Springer

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

Abstract

In this paper we study the existence of even positive homoclinic solutions for p-Laplacian ordinary differential equations (ODEs) of the type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M1">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M3">View MathML</a> and the functions a and b are strictly positive and even. First, we prove a result on symmetry of positive solutions of p-Laplacian ODEs. Then, using the mountain-pass theorem, we prove the existence of symmetric positive homoclinic solutions of the considered equations. Some examples and additional comments are given.

MSC: 34B18, 34B40, 49J40.

Keywords:
p-Laplacian ODEs; homoclinic solution; weak solution; Palais-Smale condition; mountain-pass theorem

1 Introduction and main results

In this paper we prove the existence of positive homoclinic solutions for p-Laplacian ODEs of the type

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M4">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M2">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M3">View MathML</a>. We assume that

(H) the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M7">View MathML</a> are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M8">View MathML</a> are continuously differentiable, strictly positive, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M10">View MathML</a>. Let, moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M8">View MathML</a> be even functions on ℝ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M14">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M15">View MathML</a>.

By a solution of (1), we mean a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M16">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M18">View MathML</a> and Eq. (1) holds for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M19">View MathML</a>. We are looking for positive solutions of (1) which are homoclinic, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M21">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M22">View MathML</a>.

In the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M24">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M25">View MathML</a>, similar problems are considered in [1-3] using variational methods. Note that in [2] and [3] the following second-order differential equations are considered:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M26">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M27">View MathML</a>

where a, b and c are periodic, bounded functions and a and c are positive. These equations come from a biomathematics model suggested by Austin [4] and Cronin [5]. Further results and the phase plane analysis of these equations with constant coefficients are given in [6]. Note that the periodic and homoclinic solutions of p-Laplacian ODEs are considered in [7,8].

The present work is an extension of these studies to p-Laplacian ODEs. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M28">View MathML</a> be the Sobolev space of p-integrable absolutely continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M29">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M30">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M31">View MathML</a>.

We use a variational treatment of the problem considering the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M32">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M33">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M34">View MathML</a>.

Using the well-known mountain-pass theorem, we conclude that the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M35">View MathML</a> has a nontrivial critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M36">View MathML</a>, which is a solution of the restricted problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M37">View MathML</a>

(2)

Further, we obtain uniform estimates for the solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M38">View MathML</a>, extended by 0 outside <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M39">View MathML</a>. Then, a positive homoclinic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40">View MathML</a> of (1) is found as a limit of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M38">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M42">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M43">View MathML</a>. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40">View MathML</a> is also an even function.

To obtain the property, we extend the symmetry lemma of Korman and Ouyang [9] to the p-Laplacian equations. The result is formulated and proved in Section 2.

Our main result is:

Theorem 1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M3">View MathML</a>and assumptions (H) hold. Then Eq. (1) has a positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M48">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M49">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M50">View MathML</a>. Moreover, the solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40">View MathML</a>is an even function, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M52">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M53">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M54">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M55">View MathML</a>.

Theorem 1 is proved in Section 3. From its proof we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M56">View MathML</a>

from which it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M57">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M53">View MathML</a>. Observe that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M59">View MathML</a>, the problem

has a unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M61">View MathML</a>. Indeed, multiplying the equation by u and integrating by parts over ℝ, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M62">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M63">View MathML</a>.

A simplified method can be applied to the equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M64">View MathML</a>

(3)

under assumptions (H) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M3">View MathML</a>. Note that in this case, the even homoclinic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40">View MathML</a> of Eq. (3) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M68">View MathML</a>

and again <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M57">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M53">View MathML</a>. If a and b are constants, Eq. (3) is a conservative system and one can plot the phase curves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M71">View MathML</a> in the phase plane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M72">View MathML</a>. An example is given at the end of Section 3.

2 Preliminary results

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M74">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M75">View MathML</a>. It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M76">View MathML</a> is a differentiable function and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M77">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M78">View MathML</a> exists and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M79">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M74">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M81">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M82">View MathML</a> be the space of Lebesgue measurable functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M83">View MathML</a> such that the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M84">View MathML</a>.

The dual space of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M81">View MathML</a> is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M86">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M87">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M88">View MathML</a> be the duality pairing between <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M81">View MathML</a>. By the Hölder inequality, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M91">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M92">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M93">View MathML</a>. We will use the following lemmata in further considerations.

Lemma 2For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M94">View MathML</a>, the following inequality holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M95">View MathML</a>

Proof of Lemma 2. Note that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M93">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M97">View MathML</a>. From the Hölder inequality, we have

 □

Lemma 3Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M100">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M101">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M102">View MathML</a>

The statement of Lemma 3 follows simply from the identity

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M103">View MathML</a>

The one-dimensional p-Laplacian operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M104">View MathML</a> for a differentiable function u on the interval I is introduced as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M105">View MathML</a>. Let us consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M106">View MathML</a>

(4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M107">View MathML</a> and satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M108">View MathML</a>

(5)

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M29">View MathML</a> is said to be a solution of the problem (4) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M110">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M31">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M112">View MathML</a> is absolutely continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M113">View MathML</a> holds a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M114">View MathML</a>.

We formulate an extension of Lemma 1 of [9] for p-Laplacian nonlinear equations. The result of Korman and Ouyang is one-dimensional analogue of the result of Gidas, Ni and Nirenberg [10] for symmetry of positive solutions of semilinear Laplace equations. In the case of p-Laplacian equations, the symmetry of solutions in higher dimensions is discussed by Reihel and Walter [11].

Theorem 4Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M115">View MathML</a>satisfies (5). Then any positive solutionuof (4) is an even function such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M116">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M117">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M118">View MathML</a>.

Remark 1 Let us note that if the function f satisfies (5), but u is not a positive solution of (4), then u is not necessarily an even function. A simple counter example in the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M23">View MathML</a> is the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M120">View MathML</a>

The term <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M121">View MathML</a> satisfies (5) in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M122">View MathML</a>, but the solution of the problem <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M123">View MathML</a> is negative in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M122">View MathML</a> and not an even function. Its graph is presented in Figure 1. It would be more interesting to show an example for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M125">View MathML</a> and f satisfying the additional assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M126">View MathML</a>.

thumbnailFigure 1. Graph of the functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M127">View MathML</a>.

Sketch of Proof of Theorem 4 Suppose that the function u has only one global maximum on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M128">View MathML</a>.

Assume that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M129">View MathML</a> has a finite number of local minima in the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M130">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M131">View MathML</a> be the largest local minimum. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M132">View MathML</a> be the local maximum and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M133">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M134">View MathML</a>. Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M135">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M136">View MathML</a>, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M137">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M138">View MathML</a> be the inverse functions of the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M139">View MathML</a> in the intervals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M140">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M141">View MathML</a>, respectively. Multiplying the equation in (4) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M142">View MathML</a> and integrating in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M143">View MathML</a>, we obtain by Lemma 3 and (5):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M144">View MathML</a>

which leads to contradiction. One can prove the last fact using other arguments; see, for instance, Theorem 2.1 of [12]. Suppose now that u has infinitely many local minima in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M145">View MathML</a>. Further, we can follow the steps of the proof of Lemma 1 of [9] with corresponding modifications based on Lemma 3. □

3 Proof of the main result

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M146">View MathML</a> be the Sobolev space of p-integrable absolutely continuous functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M29">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M148">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M31">View MathML</a>. Note that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M7">View MathML</a> is strictly positive and bounded, i.e., there exist a and A such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M9">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M152">View MathML</a> is an equivalent norm in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M153">View MathML</a>.

We need an extension to the p-case of the following proposition by Rabinowitz [13].

Proposition 5Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M154">View MathML</a>. Then:

(i) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M155">View MathML</a>, for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M156">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M157">View MathML</a>

(6)

(ii) For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M158">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M159">View MathML</a>

(7)

Proof of Proposition 5 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M160">View MathML</a>. It follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M161">View MathML</a>

Integrating with respect to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M162">View MathML</a> and using the Hölder and Jensen inequalities, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M163">View MathML</a>

(ii) Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M164">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M165">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M166">View MathML</a> such that by (i)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M167">View MathML</a>

 □

We are looking for positive solutions of (1), which are homoclinic, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M20">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M21">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M50">View MathML</a>. Firstly, we look for positive solutions of the problem

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M29">View MathML</a> is said to be a solution of the problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M173">View MathML</a>) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M110">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M31">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M176">View MathML</a> is absolutely continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M177">View MathML</a> holds a.e. in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M114">View MathML</a>.

A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M29">View MathML</a> is said to be a weak solution of the problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M173">View MathML</a>) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M181">View MathML</a>

Standard arguments show that a weak solution of the problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M173">View MathML</a>) is a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M173">View MathML</a>) (see [14] and [15]). Consider the modified problem

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M185">View MathML</a>. It is easy to see that solutions of the problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M186">View MathML</a>) are positive solutions of the problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M173">View MathML</a>). Indeed, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M129">View MathML</a> is a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M186">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M129">View MathML</a> has negative minimum at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M191">View MathML</a>, since for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M74">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M193">View MathML</a>, by the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M194">View MathML</a>, we reach a contradiction

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M195">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M196">View MathML</a> and u is a solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M173">View MathML</a>). We use a variational treatment of the problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M186">View MathML</a>), considering the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M32">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M200">View MathML</a>

Critical points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M201">View MathML</a> are weak solutions of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M186">View MathML</a>), i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M203">View MathML</a>

and, by a standard way, they are solutions of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M186">View MathML</a>). We show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M201">View MathML</a> satisfies the assumptions of the mountain-pass theorem of Ambrosetti and Rabinowitz [16].

Theorem 6 (Mountain-pass theorem)

LetXbe a Banach space with norm<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M206">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M207">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M208">View MathML</a>andIsatisfy the (PS) condition. Suppose that there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M209">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M210">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M211">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M212">View MathML</a>

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M213">View MathML</a>if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M214">View MathML</a>,

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M215">View MathML</a>. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M216">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M217">View MathML</a>

Thencis a critical value ofI, i.e., there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M218">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M219">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M220">View MathML</a>.

Next, denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M221">View MathML</a> several positive constants.

Lemma 7Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M3">View MathML</a>and assumptions (H) hold. Then for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M224">View MathML</a>, the problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M173">View MathML</a>) has a positive solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M38">View MathML</a>. Moreover, there is a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M227">View MathML</a>, independent ofT, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M228">View MathML</a>

(8)

ProofStep 1.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M201">View MathML</a>satisfies the (PS) condition.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M230">View MathML</a> be a sequence, and suppose there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M231">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M232">View MathML</a> such that for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M233">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M234">View MathML</a>

(9)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M235">View MathML</a>

(10)

Let us denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M236">View MathML</a>. From (9) and (10), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M237">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M238">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M239">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M240">View MathML</a>

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M241">View MathML</a>

which implies that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M242">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M153">View MathML</a>. By the compact embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M244">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M245">View MathML</a> and the subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M242">View MathML</a>, still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M242">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M248">View MathML</a> weakly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M153">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M250">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M251">View MathML</a>. We will show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M250">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M153">View MathML</a> using Lemma 2. By uniform convergence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M254">View MathML</a> to u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M255">View MathML</a>, it follows that

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M257">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M258">View MathML</a>

and by Lemma 2,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M259">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M260">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M261">View MathML</a> and by the uniform convexity of the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M153">View MathML</a>, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M263">View MathML</a>, as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M264">View MathML</a>.

Step 2. Geometric conditions.

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M265">View MathML</a>. By assumption (H) it follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M266">View MathML</a>

if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M267">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M268">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M269">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M270">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M271">View MathML</a> and also <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M272">View MathML</a>. Consider the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M273">View MathML</a>

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M274">View MathML</a>

for μ large enough.

By the mountain-pass theorem, there exists a solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M275">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M276">View MathML</a>

(11)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M277">View MathML</a>

Moreover, using the variational characterization (11), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M278">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M279">View MathML</a> is a nontrivial and positive solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M173">View MathML</a>). By Theorem 4, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M281">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M282">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M118">View MathML</a>.

Step 3. Uniform estimates.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M284">View MathML</a>. By continuation with zero of a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M245">View MathML</a> to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M286">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M287">View MathML</a>and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M288">View MathML</a>. Using the variational characterization (11), we infer that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M289">View MathML</a> and then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M290">View MathML</a>

(12)

Multiplying the equation of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M173">View MathML</a>) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M292">View MathML</a> and integrating by parts, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M293">View MathML</a>

Then by (12),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M294">View MathML</a>

We get (8) with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M295">View MathML</a>, which completes the proof. □

Proof of Theorem 1 Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M296">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M297">View MathML</a> be the solution of the problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M298">View MathML</a>) given by Lemma 2. Consider the extension of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M297">View MathML</a> to ℝ with zero outside <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M300">View MathML</a> and denote it by the same symbol.

Claim 1. The sequence of functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M301">View MathML</a>is uniformly bounded and equicontinuous.

By (8) and the embedding of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M302">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M303">View MathML</a>, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M304">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M305">View MathML</a>. Then by the equation of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M298">View MathML</a>), it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M307">View MathML</a>

(13)

By the mean value theorem for every natural n and every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M308">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M309">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M310">View MathML</a>

Then, as a consequence of (13), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M311">View MathML</a>

(14)

from which it follows <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M312">View MathML</a> and the sequence of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M313">View MathML</a> is equicontinuous. Further, we claim that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M314">View MathML</a> is also equicontinuous.

Claim 2. The sequence of functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M314">View MathML</a>is equicontinuous.

To prove this statement, we follow the method given by Tang and Xiao [7]. For completeness, we present it in details.

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M314">View MathML</a> is not an equicontinuous sequence in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M317">View MathML</a>. Then there exist an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M318">View MathML</a> and sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M319">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M320">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M321">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M322">View MathML</a>

(15)

By (14), there are numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M323','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M323">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M324">View MathML</a> and the subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M325">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M326">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M327">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M264">View MathML</a>. By (15), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M329">View MathML</a>. On the other hand, by (13) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M330">View MathML</a>

Then passing to a limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M264">View MathML</a>, we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M332','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M332">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M333">View MathML</a> which contradicts <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M329">View MathML</a>. Thus, the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M314">View MathML</a> is equicontinuous.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M224">View MathML</a>. By Claim 1 and Claim 2 and the Arzelà-Ascoli theorem, there is a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M301">View MathML</a>, still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M338">View MathML</a>, and functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M339">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M340','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M340">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M341">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M342">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M343">View MathML</a>. Trivially, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M344">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M345">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M346">View MathML</a>. Repeating this procedure as in [7], we obtain that there is a subsequence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M301">View MathML</a>, still denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M301">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M350">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M351">View MathML</a>. The function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40">View MathML</a> satisfies Eq. (1). Indeed, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M353">View MathML</a> be an interval of ℝ and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M354">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M355">View MathML</a>. By the above considerations, taking a limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M356">View MathML</a> in the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M357">View MathML</a>

equivalent to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M358">View MathML</a>

we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M359','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M359">View MathML</a>

and hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M360">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M131">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M362">View MathML</a> are arbitrary, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40">View MathML</a> is a solution of (1). Moreover, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M364">View MathML</a>

(16)

It remains to show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40">View MathML</a> is nonzero and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M366">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M367','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M367">View MathML</a>.

By Theorem 4, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M297">View MathML</a> is an even function and attains its maximum at 0. Then by Eq. (1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M369','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M369">View MathML</a>

By assumption (H)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M370','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M370">View MathML</a>

independently of n. Hence, passing to a limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M356">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M372','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M372">View MathML</a>

Note, that this implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M373','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M373">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M53">View MathML</a>.

From (16) and Proposition 5, it follows

(17)

so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M366">View MathML</a>.

Now, we will show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M377','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M377">View MathML</a>. The arguments for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M378','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M378">View MathML</a> are similar.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M379','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M379">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M380','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M380">View MathML</a> and a monotone increasing sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M381','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M381">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M382','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M382">View MathML</a>. Then for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M383','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M383">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M384','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M384">View MathML</a>

which contradicts (16).

Moreover, u is an even function that attains its only maximum at 0, since the same holds for the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M297">View MathML</a>. Arguing as in the proof of Theorem 4, we easily obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M386','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M386">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M55">View MathML</a>. □

Remark 2 A simplified method can be applied to the equations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M388','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M388">View MathML</a>

under assumptions (H) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M3">View MathML</a>. Namely, first one looks for the even positive solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M38">View MathML</a> of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M392','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M392">View MathML</a>

considering the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M393','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M393">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M394','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M394">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M395">View MathML</a> is the Sobolev space of square integrable functions such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M396','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M396">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M395','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M395">View MathML</a> is a Hilbert space, compactly embedded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M398','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M398">View MathML</a> the proof of the (PS)-condition is easier. Similar considerations are made in [1] and [3]. Then, the even homoclinic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40">View MathML</a> is obtained as a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M400','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M400">View MathML</a> limit of the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M38">View MathML</a>. Note that in this case, the even homoclinic solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M40">View MathML</a> of Eq. (3) satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M403','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M403">View MathML</a>

and again <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M57">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M53">View MathML</a>. If a and b are constants, Eq. (3) is a conservative system and one can plot the phase curves <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M71">View MathML</a> in the phase plane <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M72">View MathML</a>. Consider the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M408','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M408">View MathML</a>. The phase portrait in a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M409','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M409">View MathML</a> plane, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M410','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M410">View MathML</a> in the rectangle <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M411','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M411">View MathML</a>, is plotted on Figure 2.

thumbnailFigure 2. Phase portrait of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M412','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M412">View MathML</a>, in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M413','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/121/mathml/M413">View MathML</a>.

Competing interests

The author declares that he has no competing interests.

Acknowledgements

Dedicated to Professor Jean Mawhin on the occasion of his 70th anniversary.

The author thanks Prof. Alberto Cabada and Prof. Luis Sanchez for helpful remarks concerning Theorem 4. The author would like to thank the Department of Mathematics and Theoretical Informatics at the Technical University of Kosice, Slovakia, where the paper was prepared during his visit on the SAIA Fellowship programme. The author is thankful to the editor and anonymous referee for their comments and suggestions on the article.

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