Open Access Research

Infinitely many solutions for a boundary value problem with impulsive effects

Gabriele Bonanno1*, Beatrice Di Bella2 and Johnny Henderson3

Author Affiliations

1 Department of Civil, Information Technology, Construction, Environmental Engineering and Applied Mathematics, University of Messina, Messina, 98166, Italy

2 Department of Mathematics and Computer Science, University of Messina, Messina, 98166, Italy

3 Department of Mathematics, Baylor University, Waco, 76798-7328, USA

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Boundary Value Problems 2013, 2013:278  doi:10.1186/1687-2770-2013-278


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/278


Received:24 October 2013
Accepted:1 December 2013
Published:20 December 2013

© 2013 Bonanno et al.; licensee Springer.

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

Abstract

In this paper we are interested in multiplicity results for a nonlinear Dirichlet boundary value problem subject to perturbations of impulsive terms. The study of the problem is based on the variational methods and critical point theory. Infinitely many solutions follow from a recent variational result.

MSC: 34B37, 34B15.

Keywords:
impulsive differential equations; critical points; infinitely many solutions

1 Introduction

The theory of impulsive differential equations provides a general framework for the mathematical modeling of many real world phenomena; see, for instance, [1-3] and [4]. Indeed, many dynamical systems have an impulsive dynamical behavior due to abrupt changes at certain instants during the evolution process. Impulsive differential equations are basic tools for studying these phenomena [5,6].

There are some common techniques to approach these problems: the fixed point theorems [7,8], the method of upper and lower solutions [9], or the topological degree theory [10-12]. On the other hand, in the last few years, some authors have studied the existence of solutions by variational methods; see [13-19].

Here, we use critical point theory to investigate the existence of infinitely many solutions for the following nonlinear impulsive differential problem:

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M4">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M5">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M9">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M10">View MathML</a> are continuous for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M11">View MathML</a>.

We establish some multiplicity results for problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12">View MathML</a>) under an appropriate oscillation behavior of the primitive of the nonlinearity g and a suitable growth of the primitive of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M13">View MathML</a> at infinity, for all λ belonging to a precise interval and provided μ is small enough (Theorem 3.3, Theorem 3.4). It is worth noticing that, when the impulsive effects <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M15">View MathML</a>, are sublinear at infinity, our results hold for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M16">View MathML</a> (see Remark 3.1). Here, as an example of our results, we present the following special case of Theorem 3.3.

Theorem 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M17">View MathML</a>be a continuous function and put<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M18">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M19">View MathML</a>. Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M20">View MathML</a>

Then there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M21">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M22">View MathML</a>, such that for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M23">View MathML</a>, the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M24">View MathML</a>

admits infinitely many pairwise distinct classical solutions.

We explicitly observe that in Theorem 1.1 impulsive effects <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M15">View MathML</a>, (that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M27">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M28">View MathML</a>) are linear, contrary to the usual assumption of sublinearity of impulses; see [14,16,20-22] and [23]. The rest of this paper is organized as follows. In Section 2, we introduce some notations and preliminary results. Moreover, the abstract critical point theorem (Theorem 2.1) is recalled. In Section 3, we obtain some existence results. In Section 4, we give some examples to illustrate our results.

2 Preliminaries

By a classical solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12">View MathML</a>) we mean a function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M30">View MathML</a>

that satisfies the equation in (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12">View MathML</a>) a.e. on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M32">View MathML</a>, the limits <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M15">View MathML</a>, exist, that satisfies the impulsive conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M36">View MathML</a> and the boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M37">View MathML</a>. Clearly, if a, b and g are continuous, then a classical solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M38">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M39">View MathML</a>, satisfies the equation in (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12">View MathML</a>) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M41">View MathML</a>.

We consider the following slightly different form of problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12">View MathML</a>):

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M44">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M45">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M46">View MathML</a>.

It is easy to see that, by choosing

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M47">View MathML</a>

the solutions of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M48">View MathML</a>) are solutions of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12">View MathML</a>).

Let us introduce some notations. In the Sobolev space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M50">View MathML</a>, consider the inner product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M51">View MathML</a>

which induces the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M52">View MathML</a>

The following lemmas are useful for proving our main result. Their proofs can be found in [24].

Lemma 1 ([[24], Proposition 2.1])

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M53">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M54">View MathML</a>

(1)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M55">View MathML</a>.

Here, and in the sequel, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M56">View MathML</a> is an L1-Carathéodory function, namely:

(a) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M57">View MathML</a> is measurable for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M28">View MathML</a>;

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M59">View MathML</a> is continuous for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M60">View MathML</a>;

(c) for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M61">View MathML</a>, there exists a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M62">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M63">View MathML</a>

for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M60">View MathML</a>.

Definition 1 A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M53">View MathML</a> is said to be a weak solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M48">View MathML</a>) if u satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M67">View MathML</a>

(2)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M68">View MathML</a>.

Lemma 2 ([[24], Lemma 2.1 ])

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M53">View MathML</a>is a weak solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M48">View MathML</a>) if and only ifuis a classical solution of (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M48">View MathML</a>).

Now, we define the functionals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M72">View MathML</a> in the following way:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M73">View MathML</a>

(3)

for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M53">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M75">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M76">View MathML</a>. Using the property of f and the continuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M78">View MathML</a>, we have that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M79">View MathML</a> and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M68">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M81">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M82">View MathML</a>

So, arguing in a standard way, it is possible to prove that the critical points of the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M83">View MathML</a> are the weak solutions of problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M48">View MathML</a>) and so they are classical solutions.

In the next section we shall prove our results applying the following infinitely many critical points theorem obtained in [25]. First, we recall the following definition.

Definition 2 Let X be a real Banach space, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M85">View MathML</a> two Gâteaux differentiable functionals, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M86">View MathML</a>. We say that functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M87">View MathML</a> satisfies the Palais-Smale condition cut off upper at r (in short <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M88">View MathML</a>-condition) if any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M89">View MathML</a>, such that

(α) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M90">View MathML</a> is bounded,

(β) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M91">View MathML</a>,

(γ) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M92">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M93">View MathML</a>,

has a convergent subsequence.

When <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M94">View MathML</a>, the previous definition is the same as the classical definition of the Palais-Smale condition, while if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M95">View MathML</a>, such a condition is more general than the classical one. We refer to [25] for more details.

Theorem 2.1 (see [25], Theorem 7.4)

LetXbe a real Banach space, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M96">View MathML</a>be two continuously Gâteaux differentiable functionals such that Φ is bounded from below. For every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M97">View MathML</a>, let us put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M98">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M99">View MathML</a>

(a) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M100">View MathML</a>and for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M101">View MathML</a>, the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M102">View MathML</a>satisfies the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M103">View MathML</a>-condition for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M104">View MathML</a>, then for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M105">View MathML</a>, the following alternative holds: either

(a1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M106">View MathML</a>has a global minimum

or

(a2) there exists a sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M89">View MathML</a>of critical points (local minima) of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M106">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M109">View MathML</a>.

(b) If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M110">View MathML</a>and for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M111">View MathML</a>, the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M102">View MathML</a>satisfies the<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M103">View MathML</a>-condition for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M104">View MathML</a>, then for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M115">View MathML</a>, the following alternative holds: either

(b1) there exists a global minimum of Φ which is a local minimum of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M106">View MathML</a>

or

(b2) there exists a sequence of pairwise distinct critical points (local minima) of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M106">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M118">View MathML</a>.

We recall that Theorem 2.1 improves [[26], Theorem 2.5] since no assumptions with respect to weak topology of X are made. In particular, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M119">View MathML</a> is not involved in the definition of φ and the sequential weak lower semicontinuity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M106">View MathML</a> is not required.

3 Main results

In this section, we present our main results. Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M121">View MathML</a>

Moreover, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M122">View MathML</a>

Our first result is as follows.

Theorem 3.1Assume that

(a1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M123">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M124">View MathML</a>;

(a2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M125">View MathML</a>.

Then, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M126">View MathML</a>and for every continuous function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M15">View MathML</a>, whose potential<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M19">View MathML</a>, satisfies

(i1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M131">View MathML</a>;

(i2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M132">View MathML</a>,

there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M133">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M134">View MathML</a>

such that for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M135">View MathML</a>, problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M48">View MathML</a>) has an unbounded sequence of weak solutions.

Proof First, we observe that owing to (a2) the interval Λ is non-empty. Moreover, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M137">View MathML</a> and taking into account that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M138">View MathML</a>, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M133">View MathML</a>. Now, fix λ and μ as in the conclusion. Our aim is to apply Theorem 2.1. For this end, take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M140">View MathML</a> and Φ, Ψ as in (3).

We divide our proof into three steps in order to show Theorem 3.1. First, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M141">View MathML</a> satisfies the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M88">View MathML</a>-condition for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M104">View MathML</a>. So, fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M104">View MathML</a> and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M145">View MathML</a> be a sequence such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M146">View MathML</a> is bounded, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M147">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M92">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M149">View MathML</a>. From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M92">View MathML</a>, taking into account that Φ is coercive, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M89">View MathML</a> is bounded in X. Since the embedding of X in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M152">View MathML</a> is compact (see, for instance, [[27], Theorem 8.8]) and X is reflexive, up to a subsequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M153">View MathML</a> is uniformly convergent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M154">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M89">View MathML</a> is weakly convergent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M156">View MathML</a> in X. The uniform convergence of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M89">View MathML</a>, taking also into account Lebesgue’s theorem, ensures that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M158">View MathML</a>

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M159">View MathML</a>

(4)

Now, from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M160">View MathML</a>, there is a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M161">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M162">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M163">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M164">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M165">View MathML</a> and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M93">View MathML</a>. Setting = <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M167">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M168">View MathML</a>

(5)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M169">View MathML</a>. Moreover, having in mind that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M170">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M171">View MathML</a>

that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M172">View MathML</a>

(6)

From (5) and (6) one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M173">View MathML</a>

and owing to (4), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M174">View MathML</a>

Hence, [[27], Proposition III.30] ensures that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M89">View MathML</a> strongly converges to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M176">View MathML</a> and our claim is proved.

Second, we wish to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M177">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M178">View MathML</a> be a sequence of positive numbers such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M179">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M180">View MathML</a>

Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M181">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M93">View MathML</a>. By Lemma 1, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M183">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M184">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M164">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M186">View MathML</a>. Hence, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M187">View MathML</a>

So, from assumptions (a2) and (i2),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M188">View MathML</a>

Assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M189">View MathML</a> immediately yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M190">View MathML</a>

that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M191">View MathML</a>. The previous inequality assures that conclusion (a) of Theorem 2.1 can be used, for which either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M192">View MathML</a> has a global minimum or there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M193">View MathML</a> of solutions of problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M48">View MathML</a>) such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M195">View MathML</a>.

The final step is to verify that the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M192">View MathML</a> has no global minimum. From <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M197">View MathML</a>, and taking into account that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M198">View MathML</a>, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M199">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M200">View MathML</a>

(7)

So, there exists a sequence of positive numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M201">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M202">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M203">View MathML</a>

It follows that there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M204">View MathML</a> such that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M205">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M206">View MathML</a>

Now, consider a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M207">View MathML</a> defined by setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M208">View MathML</a>

Clearly, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M209">View MathML</a>

Moreover, bearing in mind (a1) and (i1),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M210">View MathML</a>

(8)

Putting together (7) and (8), we get that the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M192">View MathML</a> is unbounded from below and so it has no global minimum.

Therefore, Theorem 2.1 assures that there is a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M212">View MathML</a> of critical points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M192">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M214">View MathML</a> and, taking into account the considerations made in Section 2, the theorem is completely proved. □

Remark 3.1 Assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M215">View MathML</a>. Clearly, condition (a1) holds, and condition (a2) assumes the following simpler form:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M216">View MathML</a>)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M217">View MathML</a>

In particular, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M218">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M219">View MathML</a>, then (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M216">View MathML</a>) holds and problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M48">View MathML</a>) has an unbounded sequence of weak solutions in X for every pair <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M222">View MathML</a>.

Moreover, under the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M223">View MathML</a>, Theorem 3.1 guarantees the existence of infinitely many solutions to problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M48">View MathML</a>) for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M16">View MathML</a>.

As an example, we point out below a special case of Theorem 3.1.

Corollary 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M226">View MathML</a>be a continuous function, put<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M227">View MathML</a>for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M19">View MathML</a>, and let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M229">View MathML</a>. Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M230">View MathML</a>

Then, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M231">View MathML</a>, and for each continuous function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M232">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M233">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M15">View MathML</a>, the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M235">View MathML</a>

admits infinitely many pairwise distinct classical solutions.

Replacing the condition at infinity of the potential F by a similar one at zero, and arguing as in the proof of Theorem 3.1 but using conclusion (b) of Theorem 2.1 instead of (a), one establishes the following result. Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M236">View MathML</a>

Theorem 3.2Assume that

(a1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M123">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M124">View MathML</a>;

(b2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M239">View MathML</a>.

Then, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M240">View MathML</a>and for every continuous function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M11">View MathML</a>, whose potential<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M19">View MathML</a>, satisfies

(i1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M131">View MathML</a>;

(j2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M246">View MathML</a>,

there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M247">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M248">View MathML</a>

such that for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M249">View MathML</a>, problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M48">View MathML</a>) has a sequence of non-zero weak solutions, which strongly converges to 0.

Proof We take X, Φ and Ψ as in the proof of Theorem 3.1. Fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M251">View MathML</a>, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M13">View MathML</a> be a function that satisfies assumptions (i1) and (j2) and take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M253">View MathML</a>. Arguing as in the proof of Theorem 3.1, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M254">View MathML</a>. Now, arguing again as in the proof of Theorem 3.1, there is a sequence of positive numbers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M255">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M256">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M257">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M205">View MathML</a> and for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M204">View MathML</a>. By choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M260">View MathML</a> as in the proof of Theorem 3.1, the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M261','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M261">View MathML</a> strongly converges to 0 in X and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M262">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M263">View MathML</a>. Therefore, taking into account that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M264">View MathML</a>, 0 is not a local minimum of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M265">View MathML</a>. The part (b) of Theorem 2.1 ensures that there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M89">View MathML</a> in X of critical points of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M265">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M268">View MathML</a> and the proof is complete. □

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M269">View MathML</a> be a primitive of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M270">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M271">View MathML</a> an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M272">View MathML</a>-Carathéodory function and put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M273','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M273">View MathML</a>

Moreover, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M274">View MathML</a>

In virtue of Theorems 3.1 and 3.2, we obtain the following results for problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12">View MathML</a>).

Theorem 3.3Assume that

(c1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M276">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M124">View MathML</a>;

(c2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M278">View MathML</a>.

Then, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M279">View MathML</a>and for every continuous function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M78">View MathML</a>, whose potential<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M19">View MathML</a>, satisfies

(i1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M131">View MathML</a>,

(i2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M132">View MathML</a>,

there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M133">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M287">View MathML</a>

such that for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M288">View MathML</a>, problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12">View MathML</a>) has an unbounded sequence of weak solutions.

Proof As seen in Section 2, we put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M290">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M291">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M292">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M60">View MathML</a>. Clearly, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M294','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M294">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M295">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M296">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M297">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M298">View MathML</a>. Hence, from Theorem 3.1 the conclusion is achieved. □

Remark 3.2 Theorem 1.1 in Introduction is an immediate consequence of Theorem 3.3. In fact, it is enough to observe that (c1) is verified and one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M299">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M300">View MathML</a>, for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M301">View MathML</a>. Moreover, from <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M302','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M302">View MathML</a>, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M303">View MathML</a> and the conclusion is achieved.

Replacing the condition at infinity of the potential G by a similar one at zero, one establishes the following result. Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M304">View MathML</a>

Theorem 3.4Assume

(c1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M276">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M124">View MathML</a>;

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M307">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M308">View MathML</a>.

Then, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M309">View MathML</a>and for every continuous function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M311">View MathML</a>, whose potential<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M19">View MathML</a>, satisfies

(i1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M131">View MathML</a>,

(j2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M246">View MathML</a>,

there exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M247">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M317">View MathML</a>

such that for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M318">View MathML</a>, problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12">View MathML</a>) has a sequence of non-zero weak solutions, which strongly converges to 0.

Proof The conclusion follows from Theorem 3.2 by arguing as in the proof of Theorem 3.3. □

Remark 3.3 We point out that in Theorem 3.3 (as in Theorem 3.4) the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M320">View MathML</a> can be deleted provided that we assume the constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M321','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M321">View MathML</a> and the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M322">View MathML</a>.

Finally, we observe that the existence of infinitely many solutions to problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12">View MathML</a>) can be obtained from Theorem 3.3 and Theorem 3.4 even under small perturbations of the nonlinearity. As an example, we point out the following consequence of Theorem 3.3.

Corollary 3.2Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M324','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M324">View MathML</a>be an<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M272">View MathML</a>-Carathéodory function satisfying (c1) and (c2) of Theorem 3.3.

Then, for every<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M326','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M326">View MathML</a>, for every nonnegative<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M272">View MathML</a>-Carathéodory function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M328">View MathML</a>, whose potential<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M329">View MathML</a>satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M330','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M330">View MathML</a>

and for every continuous function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M127">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M11">View MathML</a>, whose potential<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M19">View MathML</a>, satisfies (i1) and (i2) of Theorem 3.3, there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M335','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M335">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M336','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M336">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M337','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M337">View MathML</a>

such that for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M338','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M338">View MathML</a>and for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M318">View MathML</a>, the problem

has an unbounded sequence of weak solutions.

Proof It is enough to apply Theorem 3.3 to the following function:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M341','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M341">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M342','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M342">View MathML</a> is fixed in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M343','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M343">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M344','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M344">View MathML</a> is fixed in Λ. In fact, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M345','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M345">View MathML</a>

(9)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M346','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M346">View MathML</a>

(10)

for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M347','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M347">View MathML</a>, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M348','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M348">View MathML</a>. Moreover, from (9) one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M349','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M349">View MathML</a> and from (10) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M350','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M350">View MathML</a>. Hence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M351','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M351">View MathML</a> and Theorem 3.3 ensures the conclusion. □

4 Applications

In many papers [13,20,22,28] and [23], the authors obtain the existence of infinitely many solutions for problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M12">View MathML</a>) while the impulsive term is supposed to be odd. The next examples provide problems that admit infinitely many solutions for which those other results cannot be applied.

Example 4.1 Consider the following boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M353','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M353">View MathML</a>

(11)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M354','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M354">View MathML</a> is the function defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M355','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M355">View MathML</a>

It is easy to see that conditions (a1), (a2), (i1) and (i2) of Theorem 3.1 hold. In particular, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M356','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M356">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M357','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M357">View MathML</a>

Then, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M358','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M358">View MathML</a> and for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M16">View MathML</a>, problem (11) has an unbounded sequence of solutions in X.

Now, we give an application of Theorem 3.4.

Example 4.2 Consider the Dirichlet problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M360','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M360">View MathML</a>

(12)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M361','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M361">View MathML</a> is the function defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M362','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M362">View MathML</a>

By a simple calculation, we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M363','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M363">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M364','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M364">View MathML</a>

Then, from Theorem 3.4, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M365','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M365">View MathML</a> and for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M366','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/278/mathml/M366">View MathML</a>, problem (12) admits a sequence of pairwise distinct classical solutions strongly converging at 0. We observe that, in this case, as direct computations show, also zero is a solution of the problem.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors have contributed equally to this research work. All authors read and approved the final manuscript.

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