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Variational approach to second-order impulsive dynamic equations on time scales

Victoria Otero-Espinar1* and Tania Pernas-Castaño12

Author affiliations

1 Departamento de Análise Matemática, Universidade de Santiago de Compostela, Santiago de Compostela, Galicia, 15782, Spain

2 Instituto de Ciencias Matemáticas (CSIC, UAM, UC3M, UCM), Madrid, 28049, Spain

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Citation and License

Boundary Value Problems 2013, 2013:119  doi:10.1186/1687-2770-2013-119


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/119


Received:27 February 2013
Accepted:23 April 2013
Published:9 May 2013

© 2013 Otero-Espinar and Pernas-Castaño; 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

The aim of this paper is to employ variational techniques and critical point theory to prove some conditions for the existence of solutions to a nonlinear impulsive dynamic equation with homogeneous Dirichlet boundary conditions. Also, we are interested in the solutions of the impulsive nonlinear problem with linear derivative dependence satisfying an impulsive condition.

MSC: 34B37, 34N05.

Keywords:
impulsive dynamic equations; second-order boundary value problem; variational techniques; critical point theory; time scales

1 Introduction

This paper is concerned with the existence of solutions of second-order impulsive dynamic equations on time scales. More precisely, we consider the following boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M1">View MathML</a>

where the impulsive points <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M2">View MathML</a> are right-dense points in an arbitrary time scale <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M3">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M4">View MathML</a>. Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M7">View MathML</a>, are continuous functions.

It is well known that the theory of impulsive dynamic equations provides a natural framework for mathematical modeling of many real world phenomena. The impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time.

Applications of impulsive dynamic equations arise in biology (biological phenomena involving thresholds), medicine (bursting rhythm models), pharmacokinetics, mechanics, engineering, chaos theory, etc. As a consequence, there has been a significant development in impulse theory in recent years. We can see some general and recent works on the theory of impulsive differential equations; see [1-9] and the references therein.

For a second-order dynamic equation, we usually consider impulses in the position and velocity. However, in the motion of spacecraft, one has to consider instantaneous impulses depending on the position that result in jump discontinuities in velocity, but with no change in position. The impulses only on the velocity occur also in impulsive mechanics. An impulsive problem with impulses in the derivative is considered in [10].

Moreover, we are interested in the solutions of the impulsive nonlinear problem in time scale with derivative dependence satisfying an impulsive condition. We can see, for example, recent works on the theory of impulsive differential equations in [1,3,6,8,11].

There have been several approaches to studying the solutions of impulsive dynamic equations on time scales, such as the method of lower and upper solutions, fixed-point theory [12-14]. Sobolev spaces of functions on time scales, which were first introduced in [15], opened a very fruitful new approach in the study of dynamic equations on time scales: the use of variational methods in the context of boundary value problems on time scales (see [16,17]) or in second-order Hamiltonian systems [18]. Moreover, the study of the existence and multiplicity of solutions for impulsive dynamic equations on time scales has also been done by means of the variational method (see, for example, [19,20]).

The aim of this paper is to use variational techniques and critical point theory to derive the existence of multiple solutions to (P); we refer the reader to [21-24] for a broad introduction to dynamic equations on time scales and to [25,26] for variational methods and critical point theory.

The paper is organized as follows. In Section 2 we gather together essential properties about Sobolev spaces on time scales proved in [15,27,28] which one needs to read this paper.

The goal of Section 3 is to exhibit the variational formulation for the impulsive Dirichlet problem. As we will see, all these problems can be understood and solved in terms of the minimization of a functional, usually related to the energy, in an appropriate space of functions. The results presented in the part where we address the linear problem are basic but crucial to revealing that a problem can be solved by finding the critical points of a functional. Moreover, we prove some sufficient conditions for the existence of at least one positive solution to (P).

To finish, in Section 4, we present an impulsive nonlinear problem with linear derivative dependence. We transform the problem into an equivalent one that has no dependence on the derivative, and then we prove that the problem has at least one solution. Also, with additional conditions in nonlinearities and impulse functions, we can show the existence of at least two solutions by using the mountain pass theorem.

2 Preliminaries

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M3">View MathML</a> be an arbitrary time scale. We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M3">View MathML</a> has the topology that it inherits from the standard topology on ℝ. Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M10">View MathML</a> are points in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M3">View MathML</a> and define the time scale interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M12">View MathML</a>. We denote <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M13">View MathML</a>.

Below we set out some results proved in [15,27] about Sobolev spaces on time scales.

Definition 2.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M14">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M16">View MathML</a>. We say that u belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M17">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M18">View MathML</a>, and there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M19">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M20">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M21">View MathML</a>

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M22">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M23">View MathML</a> is the set of all continuous functions on J such that they are Δ-differentiable on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M24">View MathML</a> and their Δ-derivatives are rd-continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M24">View MathML</a>.

Theorem 2.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M14">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M15">View MathML</a>. The set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M17">View MathML</a>is a Banach space with the norm defined for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M29">View MathML</a>as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M30">View MathML</a>

(2.1)

Moreover, the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M31">View MathML</a>is a Hilbert space with the inner product given for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M32">View MathML</a>by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M33">View MathML</a>

(2.2)

Proposition 2.1Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M14">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M15">View MathML</a>, then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M36">View MathML</a>, only dependent on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M37">View MathML</a>, such that the inequality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M38">View MathML</a>

holds for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M29">View MathML</a>, and hence the immersion<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M40">View MathML</a>is continuous.

Definition 2.2 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M41">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M15">View MathML</a>, define the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M43">View MathML</a> as the closure of the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M44">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M17">View MathML</a>. We define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M46">View MathML</a>.

The spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M47">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M48">View MathML</a> are endowed with the norm induced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M49">View MathML</a>, defined in (2.1), and the inner product induced by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M50">View MathML</a>, defined in (2.2). These spaces satisfy the following properties.

Proposition 2.2 (Poincare’s inequality)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M41">View MathML</a>be such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M15">View MathML</a>. Then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M53">View MathML</a>, only dependent on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M37">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M55">View MathML</a>

Proposition 2.3 (Corollary 3.3 in [27])

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M56">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M57">View MathML</a>

holds, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M58">View MathML</a>is the smallest positive eigenvalue of problem<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M59">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M60">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M61">View MathML</a>.

In the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M48">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M64">View MathML</a>, consider the inner product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M65">View MathML</a>

inducing the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M66">View MathML</a>.

It is the consequence of Poincare’s inequality that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M67">View MathML</a>

(2.3)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M68">View MathML</a>

(2.4)

3 Variational formulation of (P) and existence results

Firstly, to show the variational structure underlying an impulsive dynamic equation, we consider the lineal problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M69">View MathML</a>

where we consider J with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M63">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M73">View MathML</a>, are fixed constants.

Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M74">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M75">View MathML</a>. Moreover, assume that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M77">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M78">View MathML</a>.

Definition 3.1 We say that u is a classical solution of (LP) if the limits <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M79">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M80">View MathML</a> exist for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M81">View MathML</a> and it satisfies the equation on (LP) for Δ-almost everywhere (Δ-a.e.) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M82">View MathML</a>.

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M83">View MathML</a>, multiply the equation by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M84">View MathML</a> and integrate between 0 and T:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M85">View MathML</a>

Taking into account that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M86">View MathML</a> and integrating by parts, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M87">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M88">View MathML</a>

We define the bilinear form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M89">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M90">View MathML</a>

(3.1)

and the linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M91">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M92">View MathML</a>

(3.2)

Thus, the concept of weak solution for the impulsive problem (LP) is a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M56">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M94">View MathML</a> is valid for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M83">View MathML</a>.

We can prove that a defined by (3.1) and l defined by (3.2) are continuous, and, from Proposition 2.3, that a is coercive if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M96">View MathML</a>.

Consider <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M97">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M98">View MathML</a>

(3.3)

We can deduce the following regularity properties which allow us to assert that the solutions to (LP) are precisely the critical points of φ.

Lemma 3.1The following statements are valid.

1. φis differentiable at any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M56">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M100">View MathML</a>

2. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M56">View MathML</a>is a critical point ofφdefined by (3.3), thenuis a weak solution of the impulsive problem (LP).

We will use the following result in linear functional analysis, which ensures the existence of a critical point of φ.

Theorem 3.1 (Lax-Milgram theorem)

LetHbe a Hilbert space and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M102">View MathML</a>be a bounded bilinear form. Ifais coercive, i.e., there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M103">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M104">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M105">View MathML</a>, then for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M106">View MathML</a> (the conjugate space ofH) there exists a unique<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M105">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M108">View MathML</a>

Moreover, ifais also symmetric, then the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M109">View MathML</a>defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M110">View MathML</a>

attains its minimum atu.

By the Lax-Milgram theorem, we obtain the following result.

Theorem 3.2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M96">View MathML</a>then the problem (LP) has a weak solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M56">View MathML</a>for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M113">View MathML</a>. Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M114">View MathML</a>anduis a classical solution anduminimizes the functional (3.3), and hence it is a critical point ofφ.

3.1 Impulsive nonlinear problem

We consider the nonlinear Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M115">View MathML</a>

We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M96">View MathML</a>.

A weak solution of (P) is a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M117">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M118">View MathML</a>

for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M83">View MathML</a>.

We now consider the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M120">View MathML</a>

(3.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M121">View MathML</a>.

One can deduce, from the properties of H, f and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M122">View MathML</a>, the following regularity properties of φ.

Proposition 3.1The functionalφdefined by (3.4) is continuous, differentiable, and weakly lower semi-continuous. Moreover, the critical points ofφare weak solutions of (P).

Theorem 3.3Suppose thatfis bounded and that the impulsive functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M122">View MathML</a>are bounded. Then there is a critical point ofφ, and (P) has at least one solution.

Proof Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M124">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M125">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M81">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M127">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M128">View MathML</a>

Using that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M96">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M130">View MathML</a> such that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M56">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M132">View MathML</a>

Thus, using Proposition 2.1, (2.3) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M133">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M134">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M135">View MathML</a>.

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M136">View MathML</a>, and φ is coercive. Hence (Th. 1.1 of [26]), φ has a minimum, which is a critical point of φ. □

Theorem 3.4Suppose thatfis sublinear and the impulsive functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M122">View MathML</a>have sublinear growth. Then there is a critical point ofφand (P) has at least one solution.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M138">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M139">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M81">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M141">View MathML</a>

Again, using that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M96">View MathML</a>, Proposition 2.1, (2.3) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M143">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M144">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M145">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M146">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M147">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M148">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M149">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M56">View MathML</a>. □

4 Impulsive nonlinear problem with linear derivative dependence

Consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M151">View MathML</a>

where f and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M122">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M7">View MathML</a> are continuous and g is continuous and regressive.

We assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M154">View MathML</a>. Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M156">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M157">View MathML</a> is an exponential function. Note that, as g is regressive, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M158">View MathML</a> is the solution of the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M159">View MathML</a>

We transform the problem (NP) into the following equivalent form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M160">View MathML</a>

Obviously, the solutions of (NPE) are solutions of (NP). Consider the Hilbert space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M48">View MathML</a> with the inner product:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M162">View MathML</a>

and the norm induced

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M163">View MathML</a>

A weak solution of (NPE) is a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M117">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M165">View MathML</a>

Hence, a weak solution of (NP) is a critical point of the following functional:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M166">View MathML</a>

(4.1)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M167">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M168">View MathML</a>

It is evident that A is bilinear, continuous and symmetric.

Lemma 4.1 (Theorem 38.A of [29])

For the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M169">View MathML</a>withMnot empty, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M170">View MathML</a>has solutions in case the following hold:

(i) Xis a reflexive Banach space.

(ii) Mis bounded and weak sequentially closed.

(iii) φis sequentially lower semi-continuous onM.

Lemma 4.2 (Analogous to Lemma 2.2 of [5])

There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M171">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M172">View MathML</a>

Proof In fact, by Poincare’s inequality, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M173">View MathML</a>, we can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M174">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M175">View MathML</a>; if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M176">View MathML</a>, then we can take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M177">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M178">View MathML</a>. □

Lemma 4.3If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M56">View MathML</a>, then there exists a constant<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M180">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M181">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M182">View MathML</a>

Proof The result is followed by the following inequalities:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M183">View MathML</a>

 □

Lemma 4.4The functionalψdefined by (4.1) is continuous, continuously differentiable and weakly lower semi-continuous.

Theorem 4.1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M184">View MathML</a>, fand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M122">View MathML</a>are bounded, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M81">View MathML</a>, then (NP) has at least one solution.

Proof Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M187">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M188">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M7">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M190">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M56">View MathML</a>, using Lemma 4.3 and Proposition 2.3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M192">View MathML</a>

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M193">View MathML</a>, and ψ is coercive. Hence, ψ has a minimum, which is a critical point of ψ. □

We will apply the mountain pass theorem in order to obtain at least two critical points of ψ.

Suppose that X is a Banach space (in particular, a Hilbert space) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M194">View MathML</a> is differentiable and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M195">View MathML</a>. We say that ϕ satisfies the Palais-Smale condition if every bounded sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M196">View MathML</a> in the space X such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M197">View MathML</a> contains a convergent subsequence.

Theorem 4.2 (Mountain pass theorem)

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M198">View MathML</a>be such that it satisfies the Palais-Smale condition. Assume that there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M199">View MathML</a>and a bounded neighborhood Ω of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M200">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M201">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M202">View MathML</a>

Then there exists a critical point<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M203">View MathML</a>ofϕ.

Theorem 4.3Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M184">View MathML</a>, then the problem (NP) has at least two solutions if the following conditions hold:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M205">View MathML</a>) There exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M206">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M207">View MathML</a>such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M208">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M209">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M210">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M81">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M212">View MathML</a>) There exists a positive<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M213">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M214">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M215">View MathML</a>uniformly for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M216">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M217">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M81">View MathML</a>.

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M219">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M220">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M221">View MathML</a>uniformly for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M222">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M223">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M81">View MathML</a>.

Proof From (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M212">View MathML</a>), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M219">View MathML</a>) and the continuities of f and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M122">View MathML</a>, it is easy to see that for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M228">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M229">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M230">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M231">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M232">View MathML</a>

Hence, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M228">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M234">View MathML</a>, we have

(4.2)

(4.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M237">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M238">View MathML</a>.

From the condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M205">View MathML</a>), the following hold:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M240">View MathML</a>

Integrating the above two inequalities with respect to u on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M241">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M242">View MathML</a>, respectively (in this case, these are integrals on ℝ), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M243">View MathML</a>

That is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M244">View MathML</a>

Thus there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M245">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M246">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M247">View MathML</a>.

From the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M248">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M249">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M250">View MathML</a>

Hence, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M251">View MathML</a>

(4.4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M252">View MathML</a>.

Similarly, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M253">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M254">View MathML</a>

(4.5)

Firstly, we apply Lemma 4.1 to show that there exists ρ such that ψ has a local minimum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M255">View MathML</a>.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M48">View MathML</a> is a Hilbert space, it is easy to deduce that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M257">View MathML</a> is bounded and weak sequentially closed. Lemma 4.4 has shown that ψ is weak lower semi-continuous on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M257">View MathML</a> and, besides, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M48">View MathML</a> is a reflexive Banach space. So, by Lemma 4.1 we can have this <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M200">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M261">View MathML</a>.

Now we will show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M262">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M263">View MathML</a>.

In fact, from (4.2) and (4.3), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M264">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M265">View MathML</a>

We can choose

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M266">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M267">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M268">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M269">View MathML</a>. Besides, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M270">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M271">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M267">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M273">View MathML</a>. Hence, ψ has a local minimum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M274">View MathML</a>.

Next, we will show that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M275">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M276">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M277">View MathML</a>.

From (4.4) and (4.5), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M278">View MathML</a>

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M279">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M56">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M281">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M282','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M282">View MathML</a>

So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M283','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M283">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M206">View MathML</a>. Then there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M285">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M286">View MathML</a>.

Hence, for the above <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M287">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M275">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M289">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M290">View MathML</a>.

Then, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M291">View MathML</a>.

The next step is to show that ψ satisfies the Palais-Smale condition.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M292">View MathML</a> be a bounded sequence such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M293">View MathML</a>. Now we show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M294">View MathML</a> is bounded. By (4.1) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M295">View MathML</a>

(4.6)

Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M296">View MathML</a>

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M297">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M298">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M299">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M300">View MathML</a>, and that there exists a constant c such that

(4.7)

(4.8)

So, by (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M205">View MathML</a>) and (4.7), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M304">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M305">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M306">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M307">View MathML</a> are constants (independent of k).

Analogously, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M308">View MathML</a> (independent of k) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M309">View MathML</a>.

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M310">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M311">View MathML</a> is bounded, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M312">View MathML</a> is a bounded sequence.

Hence, there exists a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M313">View MathML</a> (for simplicity denoted again by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M313">View MathML</a>) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M313">View MathML</a> weakly converges to some u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M48">View MathML</a>. Then the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M313">View MathML</a> converges uniformly to u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M318">View MathML</a>.

By (4.6), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M319">View MathML</a>

So, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M320">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M321">View MathML</a> converges in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M48">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M48">View MathML</a> is a Hilbert space, and the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M324">View MathML</a> satisfies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M325">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M313">View MathML</a> converges to u, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M327">View MathML</a>. ψ satisfies the Palais-Smale condition.

Now, by Theorem 4.2, there exists a critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M203">View MathML</a>. Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M200">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M203">View MathML</a> are two critical points of ψ, and they are classical solutions of (NPE). Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M200">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M203">View MathML</a> are classical solutions of (NP). □

Example 4.1 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M333','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M333">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M334','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M334">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M335">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M336">View MathML</a>. Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M337">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M338">View MathML</a>. Consider the following boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M339','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M339">View MathML</a>

(4.9)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M130">View MathML</a> is a constant.

We can see that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M341">View MathML</a> is regressive and continuous. If we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M342">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/119/mathml/M343">View MathML</a>, by Theorem 4.3, Eq. (4.9) has at least two solutions.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally in this article. They read and approved the final manuscript.

Acknowledgements

The authors are grateful to the referees for their valuable suggestions that led to the improvement of the original manuscript. The research of V Otero-Espinar has been partially supported by Ministerio de Educación y Ciencia (Spain) and FEDER, Project MTM2010-15314.

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