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This article is part of the series A Tribute to Professor Ivan Kiguradze.

Open Access Research

Existence and multiplicity of solutions to boundary value problems for fourth-order impulsive differential equations

Alberto Cabada1* and Stepan Tersian2

Author Affiliations

1 Departamento de Análise Matemática, Facultade de Matemáticas, Universitade de Santiago de Compostela, Santiago de Compostela, Spain

2 Department of Mathematics, University of Ruse, Ruse, 7017, Bulgaria

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Boundary Value Problems 2014, 2014:105  doi:10.1186/1687-2770-2014-105


Dedicated to Professor Ivan Kiguradze for his merits in the mathematical sciences

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/105


Received:28 January 2014
Accepted:24 April 2014
Published:8 May 2014

© 2014 Cabada and 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 credited.

Abstract

In this paper we study the existence and the multiplicity of solutions for an impulsive boundary value problem for fourth-order differential equations. The notions of classical and weak solutions are introduced. Then the existence of at least one and infinitely many nonzero solutions is proved, using the minimization, the mountain-pass, and Clarke’s theorems.

MSC: 34B15, 34B37, 58E30.

Keywords:
fourth-order differential equations; impulsive conditions; weak solution; classical solution; Palais-Smale condition; mountain-pass theorem; Clarke’s theorem

1 Introduction

The theory of impulsive boundary value problems (IBVPs) became an important area of studies in recent years. IBVPs appear in mathematical models of processes with sudden changes in their states. Such processes arise in population dynamics, optimal control, pharmacology, industrial robotics, etc. For an introduction to theory of IBVPs one is referred to [1]. Some classical tools used in the study of impulsive differential equations are topological methods as fixed point theorems, monotone iterations, upper and lower solutions (see [2-4]). Recently, some authors have studied the existence of solutions of IBVPs using variational methods. The pioneering work in this direction is the paper of Nieto and O’Regan [5], where the second-order impulsive problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M1">View MathML</a>

(with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M2">View MathML</a>) is studied, using the minimization and the mountain-pass theorem. We mention also other papers for second-order impulsive equations as [6,7]. In several recent papers [8-10], fourth-order impulsive problems are considered via variational methods.

In this paper, we consider the boundary value problem for fourth-order differential equation with impulsive effects

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M3">View MathML</a>

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M4">View MathML</a>, the limits <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M6">View MathML</a> exist and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M8">View MathML</a>.

We look for solutions in the classical sense, as given in the next definition.

Definition 1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M9">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M11">View MathML</a> is said to be a classical solution of the problem (P), if u satisfies the equation a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M12">View MathML</a>, the limits <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M13">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M14">View MathML</a> exist and satisfy the impulsive conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M17">View MathML</a>, and boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M18">View MathML</a>.

Moreover, we introduce, for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M19">View MathML</a>, the following real functions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M20">View MathML</a>

(1)

To deduce the existence of solutions, we assume the following conditions:

(H1) The constant a is positive, b and c are continuous functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M21">View MathML</a> and there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M24">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M25">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M26">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M27">View MathML</a>. The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M29">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M30">View MathML</a>, are continuous functions.

(H2) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M31">View MathML</a> such that functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M17">View MathML</a>, satisfy the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M35">View MathML</a>

(2)

A simple example of functions fulfilling the last condition is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M36">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M37">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M17">View MathML</a>, are positive constants.

Note that (2) implies that there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M40">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M41">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M42">View MathML</a>

(3)

In the next section we will prove the following existence result for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M43">View MathML</a>.

Theorem 2Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M43">View MathML</a>and conditions (H1) and (H2) hold. Then the problem (P) has at least one nonzero classical solution.

Having in mind the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M45">View MathML</a>, we introduce the following condition:

(H3) There exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M46">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M47">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M17">View MathML</a> such that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M50">View MathML</a>, defined in (3), satisfy the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M51">View MathML</a>

A simple example of this new situation is given by the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M52">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M53">View MathML</a>.

The result to be proven is the following.

Theorem 3Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M45">View MathML</a>, the functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M17">View MathML</a>, are odd and conditions (H1) and (H3) hold. Then the problem (P) has infinitely many nonzero classical solutions.

If we consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M58">View MathML</a>

we introduce the following condition.

(H2′) There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M59">View MathML</a> and positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M37">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M38">View MathML</a> such that functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M32">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M49">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M50">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M17">View MathML</a>, satisfy the conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M67">View MathML</a>

(4)

A simple example now is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M69">View MathML</a>.

The obtained result is the following.

Theorem 4Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M43">View MathML</a>and conditions (H1) and (H2′) hold. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M71">View MathML</a>, the problem (P1) has only the zero solution. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M72">View MathML</a>, the problem (P1) has at least one nonzero classical solution.

The proofs of the main results are given in Section 3.

2 Preliminaries

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M73">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M74">View MathML</a>, the Lebesgue space of p-integrable functions over the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M75">View MathML</a>, endowed with the usual norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M76">View MathML</a>, and by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M77">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M78">View MathML</a> the corresponding norms in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M79">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M80">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M81">View MathML</a>

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M82">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M83">View MathML</a> the Sobolev spaces

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M84">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M85">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M86">View MathML</a> be the Hilbert space endowed with the usual scalar product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M87">View MathML</a>

and the corresponding norm.

By assumption (H1) an equivalent scalar product and norm in X are given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M88">View MathML</a>

(5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M89">View MathML</a>

(6)

It is well known (see [[9], Lemma 2.2], [11]) that the following Poincaré and imbedding inequalities hold for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M90">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M91">View MathML</a>

(7)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M92">View MathML</a>

(8)

where M is a positive constant depending on T, a and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M93">View MathML</a>.

We have the following compactness embedding, which can be proved in the standard way.

Proposition 5The inclusion<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M94">View MathML</a>is compact.

We define the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M95">View MathML</a>, as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M96">View MathML</a>

(9)

By assumption (H1), we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M95">View MathML</a> is continuously differentiable and, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M98">View MathML</a>, the following identity holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M99">View MathML</a>

(10)

In the sequel we introduce the concept of a weak solution of our problem.

Definition 6 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M90">View MathML</a> is said to be a weak solution of the problem (P), if for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M98">View MathML</a>, the following identity holds:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M102">View MathML</a>

(11)

As a consequence, the critical points of ϕ are the weak solutions of the problem (P). Let us see that they are, actually, strong solutions too.

Lemma 7Ifuis a weak solution of (P) then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M90">View MathML</a>is classical solution of (P).

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M90">View MathML</a> be a weak solution of (P), i.e. (11) holds for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M98">View MathML</a>. For a fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M106">View MathML</a> we take a test function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M107">View MathML</a>, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M108">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M109">View MathML</a>. We have by (11)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M110">View MathML</a>

This means that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M111">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M112">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M113">View MathML</a> satisfies the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M114">View MathML</a>

By a standard regularity argument (see [9,11]) the weak derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M115">View MathML</a> and therefore the limits <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M116">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M117">View MathML</a> exist.

We have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M98">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M119">View MathML</a>

Summing the last identities for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M11">View MathML</a> we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M121">View MathML</a>

(12)

Therefore, by (11) and (12), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M122">View MathML</a>

(13)

Now, take a test function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M123">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M124">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M125">View MathML</a>

Then we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M126">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M127">View MathML</a>. Similarly, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M128">View MathML</a>, which shows that u is a classical solution of the problem (P). The lemma is proved. □

In the proofs of the theorems, we will use three critical point theorems which are the main tools to obtain weak solutions of the considered problems.

To this end, we introduce classical notations and results. Let E be a reflexive real Banach space. Recall that a functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M129">View MathML</a> is lower semi-continuous (resp. weakly lower semi-continuous (w.l.s.c.)) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M130">View MathML</a> (resp. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M131">View MathML</a>) in E implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M132">View MathML</a> (see [12], pp.3-5).

We have the following well-known minimization result.

Theorem 8LetIbe a weakly lower semi-continuous operator that has a bounded minimizing sequence on a reflexive real Banach spaceE. ThenIhas a minimum<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M133">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M129">View MathML</a>is a differentiable functional, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M135">View MathML</a>is a critical point ofI.

Note that a functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M129">View MathML</a> is w.l.s.c. on I if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M137">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M138">View MathML</a> is convex and continuous and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M139">View MathML</a> is sequentially weakly continuous (i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M131">View MathML</a> in E implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M141">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M142">View MathML</a>) (see [13], pp.301-303). The existence of a bounded minimizing sequence appears, when the functional I is coercive, i.e.<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M143">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M144">View MathML</a>.

Next, recall the notion of the Palais-Smale (PS) condition, the mountain-pass theorem and Clarke’s theorem.

We say that I satisfies condition (PS) if any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M145">View MathML</a> for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M146">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M147">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M148">View MathML</a> possesses a convergent subsequence.

Theorem 9 ([[14], p.4])

LetEbe a real Banach space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M149">View MathML</a>satisfying condition (PS). Suppose<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M150">View MathML</a>and

(i) there are constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M151">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M152">View MathML</a>if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M153">View MathML</a>,

(ii) there is an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M154">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M155">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M156">View MathML</a>.

ThenIpossesses a critical value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M157">View MathML</a>. Moreover, ccan be characterized as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M158">View MathML</a>where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M159">View MathML</a>.

Theorem 10 ([[14], p.53])

LetEbe a real Banach space and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M149">View MathML</a>withIeven, bounded from below, and satisfying condition (PS). Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M150">View MathML</a>, there is a set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M162">View MathML</a>such thatKis homeomorphic to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M163">View MathML</a>by an odd map, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M164">View MathML</a>.

ThenIpossesses, at least, mdistinct pairs of critical points.

3 Proofs of main results

This section is devoted to the proof of the three theorems enunciated in the introduction of this work.

First consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M43">View MathML</a> for which we prove that the functional ϕ satisfies the Palais-Smale condition.

Lemma 11Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M43">View MathML</a>and conditions (H1) and (H2) hold. Then the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M95">View MathML</a>satisfies condition (PS).

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M168">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M169">View MathML</a> be such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M170">View MathML</a>

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M171">View MathML</a>

(14)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M172">View MathML</a>

(15)

for all sufficiently large k, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M173">View MathML</a>. Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M174">View MathML</a> in (15), we have for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M173">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M176">View MathML</a>

In particular,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M177">View MathML</a>

Adding the last inequality with (14), by assumption (H2), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M178">View MathML</a>

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M179">View MathML</a> is a bounded sequence in X.

Then, by the compact inclusion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M94">View MathML</a>, it follows that, up to a subsequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M131">View MathML</a> weakly in X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M130">View MathML</a> strongly in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M183">View MathML</a>. As a consequence, from the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M184">View MathML</a>

it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M185">View MathML</a>

(16)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M186">View MathML</a>

Then by (16) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M187">View MathML</a>

i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M130">View MathML</a> strongly in X, which completes the proof. □

Now, we are in a position to prove the main results of this paper.

Proof of Theorem 2 We find by (H1) and (8) that the following inequalities are valid for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M189">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M190">View MathML</a>

It is evident that this last expression is strictly positive when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M191">View MathML</a>, with ρ small enough. Next, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M192">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M193">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M194">View MathML</a>. Then, by (H2) and (3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M195">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M196">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M43">View MathML</a>, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M198">View MathML</a> for sufficiently large λ. According to the mountain-pass Theorem 9, together with Lemmas 11 and 7, we deduce that there exists a nonzero classical solution of the problem (P). □

Now consider the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M45">View MathML</a>. In the next result we prove that the Palais-Smale condition is also valid.

Lemma 12Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M45">View MathML</a>and conditions (H1) and (H3) hold. Then the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M95">View MathML</a>is bounded from below and satisfies condition (PS).

Proof By <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M45">View MathML</a>, conditions (H1), (H3), and inequality (8), it follows that the functional ϕ is bounded from below:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M203">View MathML</a>

(17)

Further, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M179">View MathML</a> is a (PS) sequence, by (17) it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M179">View MathML</a> is a bounded sequence in X. Then, as in Lemma 11, we conclude that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M179">View MathML</a> has a convergent subsequence. □

Now we are in a position to prove the next existence result for the problem (P).

Proof of Theorem 3 By assumption, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M32">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M33">View MathML</a> are odd functions. So <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M49">View MathML</a> are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M50">View MathML</a> are even functions and the functional ϕ is even. By Lemma 12 we know that ϕ is bounded from below and satisfies condition (PS). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M211">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M212">View MathML</a> be a natural number and define, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M213">View MathML</a> fixed, the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M214">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M215">View MathML</a> is homeomorphic to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M163">View MathML</a> by the odd mapping defined as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M217">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M218">View MathML</a>

Moreover, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M219">View MathML</a>, the following inequalities hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M220">View MathML</a>

(18)

Clearly <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M215">View MathML</a> is a subset of the m-dimensional subspace

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M222">View MathML</a>

and there exist positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M223">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M224">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M225">View MathML</a>

(19)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M226">View MathML</a> is the induced norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M227">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M228">View MathML</a>.

Arguing as in [[15], pp.16-18], one can prove that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M229">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M230">View MathML</a>

(20)

Denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M231">View MathML</a>

By (H3) we see that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M233">View MathML</a>, the following inequalities are fulfilled:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M234">View MathML</a>

(21)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M235">View MathML</a>

Denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M236">View MathML</a>. Then by (18)-(21) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M237">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M238">View MathML</a>.

By the last inequality, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M239">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M240">View MathML</a>. Then, by (18), choosing

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M241">View MathML</a>

we obtain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M239">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M232">View MathML</a>.

By Clarke’s Theorem 10, there exist at least m pairs of different critical points of the functional ϕ. Since m is arbitrary, there exist infinitely many solutions of the problem (P), which concludes the proof. □

Concerning the problem (P1), one can introduce similarly the notions of classical and weak solutions. In this case it is not difficult to verify that the weak solutions are critical points of the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M244">View MathML</a> defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M245">View MathML</a>

(22)

Proof of Theorem 4 By the Poincaré inequalities (7) we find that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M246">View MathML</a> is an equivalent norm to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M247">View MathML</a> in X and the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M248">View MathML</a> is convex.

Since the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M249">View MathML</a>

is sequentially weakly continuous, from the fact that the inclusion <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M250">View MathML</a> is compact, we deduce that the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M251">View MathML</a> is weakly lower semi-continuous.

Next, let us see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M244">View MathML</a> is bounded from below:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M253">View MathML</a>

(23)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M254">View MathML</a>

Then, by Theorem 8, there exists a minimizer of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M255">View MathML</a>, which is a critical point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M255">View MathML</a>.

Let u be a weak solution of (P1), i.e., a critical point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M255">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M258">View MathML</a>

(24)

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M71">View MathML</a> then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M260">View MathML</a>. Suppose that u is a nonzero solution and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M261">View MathML</a>. By (H2′), (7), and (24) it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M262">View MathML</a>

which is a contradiction. Then, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M261">View MathML</a>, the problem (P1) has only the zero solution.

Suppose now that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M264">View MathML</a>.

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M265">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M266">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M267">View MathML</a>

(25)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M268">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M269">View MathML</a> it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M270">View MathML</a>. Then, since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M59">View MathML</a>, by (25) it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M272">View MathML</a> for sufficiently small <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M266">View MathML</a>. In consequence we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M274">View MathML</a>. So we ensure the existence of a nonzero minimizer of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/105/mathml/M255">View MathML</a>, which completes the proof of Theorem 4. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Acknowledgements

The authors are thankful to the anonymous referees for the careful reading of the manuscript and suggestions.

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