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Ni-Serrin type equations arising from capillarity phenomena with non-standard growth

Mustafa Avci

Author Affiliations

Department of Mathematics, Faculty of Science, Dicle University, Diyarbakir, 21280, Turkey

Boundary Value Problems 2013, 2013:55  doi:10.1186/1687-2770-2013-55

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/55


Received:28 December 2012
Accepted:28 February 2013
Published:18 March 2013

© 2013 Avci; 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 the present paper, in view of the variational approach, we discuss a Ni-Serrin type equation involving non-standard growth condition and arising from the capillarity phenomena. Establishing some suitable conditions, we prove the existence and multiplicity of solutions.

MSC: 35D05, 35J60, 35J70.

Keywords:
<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M1">View MathML</a>-Laplacian; variable exponent Sobolev space; mountain pass theorem; genus theory; variational method; capillarity phenomena

1 Introduction

We study the existence and multiplicity of solutions for a Ni-Serrin type equation involving non-standard growth condition and arising from capillarity phenomena of the following type:

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M3">View MathML</a> is a bounded domain with smooth boundary Ω, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M4">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M5">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M7">View MathML</a>.

Capillarity can be briefly explained by considering the effects of two opposing forces: adhesion, i.e., the attractive (or repulsive) force between the molecules of the liquid and those of the container; and cohesion, i.e., the attractive force between the molecules of the liquid. The study of capillary phenomena has gained some attention recently. This increasing interest is motivated not only by fascination in naturally-occurring phenomena such as motion of drops, bubbles and waves but also its importance in applied fields ranging from industrial and biomedical and pharmaceutical to microfluidic systems.

The study of ground states for equations of the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M8">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M9">View MathML</a> is the Kirchhoff stress term and the source term f was very general, was initiated by Ni and Serrin [1,2]. Moreover, radial solutions of the problem (1.1) have been studied in the context of the analysis of capillarity surfaces for a function of the form <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M11">View MathML</a> (see [3-5]). Recently, in [6] Rodrigues studied a version of the problem (P) for the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M14">View MathML</a>.

We note that if we choose the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M15">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M16">View MathML</a> in (P), then we get the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M17">View MathML</a>

(1.2)

which is called the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M1">View MathML</a>-Kirchhoff type equation [7-9]. In this case, the problem (1.2) indicates a generalization of a model, the so-called Kirchhoff equation, introduced by Kirchhoff in [10]. To be more precise, Kirchhoff established a model given by the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M19">View MathML</a>

(1.3)

where ρ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M20">View MathML</a>, h, E, l are constants, which extends the classical D’Alambert wave equation by considering the effects of the changes in the length of the strings during the vibrations. A distinguishing feature of Kirchhoff equation (1.3) is that the equation contains a nonlocal coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M21">View MathML</a> which depends on the average <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M22">View MathML</a> of the kinetic energy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M23">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M24">View MathML</a>, and hence the equation is no longer a pointwise identity.

The nonlinear problems involving the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M1">View MathML</a>-Laplacian operator, that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M26">View MathML</a>, are extremely attractive because they can be used to model dynamical phenomena which arise from the study of electrorheological fluids or elastic mechanics, in the modeling of stationary thermo-rheological viscous flows of non-Newtonian fluids and in the mathematical description of the processes filtration of an ideal barotropic gas through a porous medium [11-15]. The detailed application backgrounds of the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M1">View MathML</a>-Laplacian can be found in [16-20] and references therein.

2 Abstract framework and preliminary results

We state some basic properties of the variable exponent Lebesgue-Sobolev spaces <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M28">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M29">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M3">View MathML</a> is a bounded domain (for details, see [21-24]).

Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M31">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M32">View MathML</a> and denote

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M33">View MathML</a>

For any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M32">View MathML</a>, we define the variable exponent Lebesgue space by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M35">View MathML</a>

then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M28">View MathML</a> endowed with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M37">View MathML</a>

becomes a Banach space.

Proposition 1[22,24]

For any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M38">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M39">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M40">View MathML</a>

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M41">View MathML</a>is a conjugate space of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M42">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M43">View MathML</a>.

The modular of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M28">View MathML</a>, which is the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M45">View MathML</a>, is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M46">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M47">View MathML</a>.

Proposition 2[22,24]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M48">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M49">View MathML</a>), then the following statements are equivalent:

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M50">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M51">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M52">View MathML</a>in measure in Ω and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M53">View MathML</a>.

Proposition 3[22,24]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M48">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M49">View MathML</a>), we have

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M56">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M57">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M58">View MathML</a>;

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M59">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M60">View MathML</a>.

The variable exponent Sobolev space<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M61">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M62">View MathML</a>

with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M63">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M64">View MathML</a>.

The space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M65">View MathML</a> is defined as the closure of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M66">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M61">View MathML</a> with respect to the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M68">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M69">View MathML</a>, we can define an equivalent norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M70">View MathML</a>

since the Poincaré inequality holds, i.e., there exists a positive constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M71">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M72">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M69">View MathML</a>[18,24].

Proposition 4[22,24]

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M74">View MathML</a>, then the spaces<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M61">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M65">View MathML</a>are separable and reflexive Banach spaces.

Proposition 5[22,24]

Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M78">View MathML</a>. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M79">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M80">View MathML</a>, then the embedding<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M81">View MathML</a>is compact and continuous, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M82">View MathML</a>if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M83">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M84">View MathML</a>if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M85">View MathML</a>.

Proposition 6[18]

LetXbe a Banach space and let define the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M86">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M87">View MathML</a>is convex. The mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M88">View MathML</a>is a strictly monotone, bounded homeomorphism of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M89">View MathML</a>type, namely

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M90">View MathML</a>

Definition 7 Let X be a Banach space and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M91">View MathML</a> be a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M92">View MathML</a>-functional. We say that a functional J satisfies the Palais-Smale condition ((PS) for short) if any sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M93">View MathML</a> in X, such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M94">View MathML</a> is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M95">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M96">View MathML</a>, admits a convergent subsequence.

We say that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M69">View MathML</a> is a weak solution of (P) if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M98">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M99">View MathML</a>. The energy functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M100">View MathML</a> corresponding to the problem (P) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M101">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M102">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M103">View MathML</a>.

Thanks to the conditions (M0) and (f0) (see below), the functional I is well defined and of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M92">View MathML</a>. Since the problem (P) is in the variational setting, the critical points of I are weak solutions of (P). Moreover, the derivative of I is the mapping <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M105">View MathML</a> given by the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M106">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M107">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M108">View MathML</a>

3 Main results

Theorem 8Assume the following conditions hold:

(M0) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M109">View MathML</a>is a continuous function and satisfies the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M110">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M111">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M112">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M113">View MathML</a>are positive real numbers;

(f0) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M114">View MathML</a>satisfies the Carathéodory condition and there exist positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M115">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M116">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M117">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M119">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M120">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M121">View MathML</a>. Then (P) has a weak solution.

Proof By the assumptions (M0) and (f0), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M122">View MathML</a>

Therefore, by Proposition 3 and Proposition 5, it follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M123">View MathML</a>

(3.1)

By the assumption <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M124">View MathML</a>, I is coercive. Since I is weakly lower semicontinuous, I has a minimum point u in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M65">View MathML</a> and u is a weak solution of (P). □

Theorem 9Assume the following conditions hold:

(M1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M126">View MathML</a>is a continuous function and satisfies the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M127">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M111">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M129">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M130">View MathML</a>andαreal numbers such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M131">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M113">View MathML</a>;

(M2) Msatisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M133">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M111">View MathML</a>;

(f1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M114">View MathML</a>satisfies the Carathéodory condition and there exist positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M136">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M137">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M138">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M119">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M141">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M142">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M143">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M144">View MathML</a>;

(f2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M145">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M146">View MathML</a>uniformly for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M80">View MathML</a>;

(f3) There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M148">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M149">View MathML</a>for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6">View MathML</a>and all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M151">View MathML</a>;

(AR) Ambrosetti-Rabinowitz’s condition holds, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M153">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M154">View MathML</a>

Then (P) has at least one nontrivial weak solution.

To obtain the result of Theorem 9, we need to show that Lemma 10 and Lemma 11 hold.

Lemma 10Suppose (M1), (M2), (AR) and (f1) hold. ThenIsatisfies the (PS) condition.

Proof Let us assume that there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M155">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M65">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M157">View MathML</a>

(3.2)

Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M158">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M159">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M160">View MathML</a>. Therefore,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M161">View MathML</a>

By the above inequalities and assumptions (M1), (M2) and (AR), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M162">View MathML</a>

This implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M163">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M65">View MathML</a>. Passing to a subsequence if necessary, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M69">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M166">View MathML</a>. Therefore, by Proposition 5, we have

(3.3)

By (3.2), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M168">View MathML</a>. Thus

From (f1) and Proposition 1, it follows

If we consider the relations given in (3.3), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M171">View MathML</a>

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M172">View MathML</a>

From (M1), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M173">View MathML</a>

(3.4)

Since the functional (3.4) is of type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M174">View MathML</a> (see Proposition 3.1 in [6]), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M175">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M176">View MathML</a>. We are done. □

Lemma 11Suppose (M1), (AR) and (f1)-(f3) hold. Then the following statements hold:

(i) There exist two positive real numbersγandasuch that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M69">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M179">View MathML</a>;

(ii) There exists<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M180">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M181">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M182">View MathML</a>.

Proof (i) Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M183">View MathML</a>. Then by (M1) and Proposition 3, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M184">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M185">View MathML</a>, by Proposition 5 we have the continuous embeddings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M186">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M187">View MathML</a>, and also there are positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M188">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M189">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M190">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M191">View MathML</a>

(3.5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M192">View MathML</a>

(3.6)

From (f1) and (f2), we get <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M193">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M119">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M196">View MathML</a> is small enough and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M197">View MathML</a>. Therefore, by (M1), Proposition 3 and (3.5), (3.6), it follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M198">View MathML</a>

providing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M199">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M183">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M144">View MathML</a>, there exist two positive real numbers γ and a such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M177">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M180">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M204">View MathML</a>.

(ii) From (AR) and (f3), one easily deduces

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M205">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M151">View MathML</a>. Therefore, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M208">View MathML</a> and nonnegative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M69">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M210">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M211">View MathML</a>

(recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M212">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M213">View MathML</a> almost everywhere). On the other hand, when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M214">View MathML</a>, from (M1) we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M215">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M216">View MathML</a>, it is obvious <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M217">View MathML</a>. Hence, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M218">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M219">View MathML</a>

From the assumption on θ (see (AR)), we conclude <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M220">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M221">View MathML</a>. □

Proof of Theorem 9 From Lemma 10, Lemma 11 and the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M222">View MathML</a>, I satisfies the mountain pass theorem (see [25,26]). Therefore, I has at least one nontrivial weak solution. The proof of Theorem 9 is completed. □

In the sequel, using Krasnoselskii’s genus theory (see [25,27]), we show the existence of infinitely many solutions of the problem (P). So, we recall some basic notations of Krasnoselskii’s genus.

Let X be a real Banach space and set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M223">View MathML</a>

Definition 12 Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M224">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M225">View MathML</a>. The genus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M226">View MathML</a> of E is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M227">View MathML</a>

If such a mapping does not exist for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M11">View MathML</a>, we set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M229">View MathML</a>. Note also that if E is a subset which consists of finitely many pairs of points, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M230">View MathML</a>. Moreover, from the definition, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M231">View MathML</a>. A typical example of a set of genus k is a set which is homeomorphic to a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M232">View MathML</a> dimensional sphere via an odd map.

Now, we will give some results of Krasnoselskii’s genus which are necessary throughout the present paper.

Theorem 13Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M233">View MathML</a>andΩ be the boundary of an open, symmetric and bounded subset<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M3">View MathML</a>with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M235">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M236">View MathML</a>.

Corollary 14<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M237">View MathML</a>.

Remark 15 If X is of an infinite dimension and separable and S is the unit sphere in X, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M238">View MathML</a>.

Theorem 16Suppose thatMandfsatisfy the following conditions:

(M3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M239">View MathML</a>is a continuous function and satisfies the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M240">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M111">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M242">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M243">View MathML</a>, δandαare real numbers such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M244">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M245">View MathML</a>;

(f4) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M114">View MathML</a>is a continuous function and there exist positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M247">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M248">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M249">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M250">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M251">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M253">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M254">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M255">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6">View MathML</a>;

(f5) fis an odd function according tot, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M257">View MathML</a>

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M259">View MathML</a>.

If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M260','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M260">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M262">View MathML</a>, then the problem (P) has infinitely many solutions.

The following result obtained by Clarke in [28] is the main idea which we use in the proof of Theorem 16.

Theorem 17Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M263">View MathML</a>be a functional satisfying the (PS) condition. Furthermore, let us suppose that:

(i) Jis bounded from below and even;

(ii) There is a compact set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M264','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M264">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M265">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M266">View MathML</a>.

ThenJpossesses at leastkpairs of distinct critical points and their corresponding critical values are less than<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M267">View MathML</a>.

Lemma 18Suppose (M3), (f4) and the inequality<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M262">View MathML</a>hold.

(i) Iis bounded from below;

(ii) Isatisfies the (PS) condition.

Proof (i) By the assumptions (M3) and (f4), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M269">View MathML</a>

By Proposition 3 and Proposition 5, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M270">View MathML</a>

(3.7)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M271">View MathML</a> large enough. Hence, I is bounded from below.

(ii) Let us assume that there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M272">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M65">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M274">View MathML</a>

(3.8)

From (3.8) we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M275">View MathML</a>. This fact combined with (3.7) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M276','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M276">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M277">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M262','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M262">View MathML</a>, we obtain that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M279">View MathML</a> is bounded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M65">View MathML</a>.

Hence, we may extract a subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M281','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M281">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M69">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M166">View MathML</a> in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M65">View MathML</a>. In the rest of the proof, if we consider similar relations given in (3.3) and growth conditions assumed on f and apply the same processes which we used in the proof of Lemma 10, we can see that I satisfies the (PS) condition. □

Proof of Theorem 16 Set (see [7,25])

then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M286','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M286">View MathML</a>

Now, we will show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M287','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M287">View MathML</a> for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M288">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M289">View MathML</a> is a reflexive and separable Banach space, for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M288">View MathML</a>, we can choose a k-dimensional linear subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M291">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M289">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M293','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M293">View MathML</a>. As the norms on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M291','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M291">View MathML</a> are equivalent, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M295','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M295">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M296','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M296">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M297','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M297">View MathML</a> implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M298','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M298">View MathML</a>.

Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M299','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M299">View MathML</a>. By the compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M300','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M300">View MathML</a> and the condition (f4), there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M301','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M301">View MathML</a> such that

(3.9)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M303','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M303">View MathML</a>. If we consider (M3) and (f4), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M304','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M304">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M305','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M305">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M306','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M306">View MathML</a>

(3.10)

providing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M307','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M307">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M308','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M308">View MathML</a>, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M309">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M310">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M311','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M311">View MathML</a>

i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M312','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M312">View MathML</a>

It is clear that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M313">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M314','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M314">View MathML</a>. Finally, by Lemma 18 above, we can apply Theorem 17 to obtain that the functional I admits at least k pairs of distinct critical points, and since k is arbitrary, we obtain infinitely many critical points of I. The proof is completed. □

Theorem 19Suppose (M3), (f4) and (f5) hold. If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M315','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M315">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M6">View MathML</a>, then the problem (P) has a sequence of solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M317','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M317">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M318">View MathML</a>.

Proof In the beginning, we will show that I is coercive. If we follow the same processes applied in the proof of Theorem 8 and consider the fact <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M319','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M319">View MathML</a>, it is easy to get the coerciveness of I. Since I is weak lower semi-continuous, I attains its minimum on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M320','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M320">View MathML</a>, i.e., (P) has a solution. By help of coerciveness, we know that I satisfies the (PS) condition on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M289">View MathML</a>. Moreover, from the condition (f5), I is even.

In the rest of the proof, since we develop the same arguments which we used in the proof of Theorem 16, we omit the details. Therefore, if we follow similar steps to those in (3.9) and (3.10) and consider the inequalities <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M322','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M322">View MathML</a>, we can find <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M309','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M309">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M310','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M310">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M325','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M325">View MathML</a>

Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M313','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M313">View MathML</a>, so <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M327','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M327">View MathML</a>. By Krasnoselskii’s genus, each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M328','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M328">View MathML</a> is a critical value of I, hence there is a sequence of solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M329','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M329">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M318','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/55/mathml/M318">View MathML</a>. □

Competing interests

The author declares that he has no competing interests.

Acknowledgements

The author would like to thank the referee for some valuable comments and helpful suggestions.

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