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

Open Access Research

Ground state and mountain pass solutions for discrete p ( ) -Laplacian

Cristian Bereanu1*, Petru Jebelean2 and Călin Şerban2

Author Affiliations

1 Institute of Mathematics “Simion Stoilow”, Romanian Academy, 21, Calea Griviţei, Sector 1, Bucharest, RO-010702, Romania

2 Department of Mathematics, West University of Timişoara, 4, Blvd. V. Pârvan, Timişoara, RO-300223, Romania

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Boundary Value Problems 2012, 2012:104  doi:10.1186/1687-2770-2012-104

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/104


Received:25 July 2012
Accepted:5 September 2012
Published:19 September 2012

© 2012 Bereanu 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 study the existence of solutions for discrete <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M3">View MathML</a>-Laplacian equations subjected to a potential type boundary condition. Our approach relies on Szulkin’s critical point theory and enables us to obtain the existence of ground state as well as mountain pass type solutions.

MSC: 39A12, 39A70, 49J40, 65Q10.

Keywords:
discrete <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M3">View MathML</a>-Laplacian operator; variational methods; critical point; Palais-Smale condition; Mountain Pass Theorem

1 Introduction

Let T be a positive integer, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M5">View MathML</a> be defined by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M6">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M8">View MathML</a>. Here and below, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M9">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M10">View MathML</a>, we use the notation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M11">View MathML</a>.

This paper is concerned with the existence of solutions for equations of the type

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M12">View MathML</a>

(1.1)

subjected to the potential boundary condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M13">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M14">View MathML</a> is the forward difference operator and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M15">View MathML</a> stands for the discrete <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M16">View MathML</a>-Laplacian operator, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M17">View MathML</a>

Here and hereafter, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M18">View MathML</a> is a continuous function, while <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M19">View MathML</a> is convex, proper (i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M20">View MathML</a>), lower semicontinuous (in short, l.s.c.) and ∂j denotes the subdifferential of j. Recall, for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M21">View MathML</a>, the set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M22">View MathML</a> is defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M23">View MathML</a>

(1.3)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M24">View MathML</a> stands for the usual inner product in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M25">View MathML</a>.

It should be noticed that the boundary condition (1.2) recovers the classical ones. For instance, denoting by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M26">View MathML</a> the indicator function of a closed, nonempty and convex set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M27">View MathML</a>, the Dirichlet and Neumann boundary conditions are obtained by choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M28">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M29">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M30">View MathML</a>, respectively. If p is T-periodic, taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M31">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M32">View MathML</a>) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M28">View MathML</a>, we get the periodic (antiperiodic) conditions. For other choices of j yielding various boundary conditions, we refer the reader to Gasinski and Papageorgiou [1] and Jebelean [2].

The study of boundary value problems with a discrete p-Laplacian using variational approaches has captured attention in the last years. Most of the papers deal with classical boundary conditions such as Dirichlet (see, e.g., Agarwal et al.[3], Cabada et al.[4]), Neumann (Candito and D’Agui [5], Tian and Ge [6]) and periodic (He and Chen [7], Jebelean and Şerban [8]). Also, we note the recent paper of Mawhin [9] where variational techniques are employed to obtain the existence of periodic solutions for systems involving a general discrete ϕ-Laplacian operator.

Boundary value problems with the discrete <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M16">View MathML</a>-Laplacian subjected to Dirichlet, Neumann or periodic boundary conditions were studied in recent time by Bereanu et al.[10], Galewski and Glab [11,12], Guiro et al.[13], Koné and Ouaro [14], Mashiyev et al.[15], Mihăilescu et al.[16,17].

Here, we use a variational approach to obtain ground state and mountain pass solutions for problem (1.1), (1.2). In this view, we employ some ideas originated in Jebelean and Moroşanu [18] (also see Jebelean [2]) combined with specific technicalities due to the discrete and anisotropic character of the problem. The main existence results are Theorem 3.1 and Theorem 4.2. These recover and generalize the similar ones for p= constant obtained in [19].

The rest of the paper is organized as follows. The functional framework and the variational approach of problem (1.1), (1.2) are presented in Section 2. In Section 3, we obtain the existence of ground state solutions, while Section 4 is devoted to the existence of mountain pass type solutions. An example of application is given in Section 5.

2 The functional framework

Our approach for the boundary value problem (1.1), (1.2) relies on the critical point theory developed by Szulkin [20]. With this aim, we introduce the space

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M35">View MathML</a>

which will be considered with the Luxemburg norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M36">View MathML</a>

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M37">View MathML</a>. Also, we shall make use of the usual sup-norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M38">View MathML</a>.

Next, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M39">View MathML</a> be defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M40">View MathML</a>

(2.1)

Standard arguments show that φ is convex, of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M41">View MathML</a>. Using the summation by parts formula (see, e.g., [8,19]), one obtains that its derivative is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M42">View MathML</a>

(2.2)

By means of j, we introduce the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M43">View MathML</a> given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M44">View MathML</a>

(2.3)

Note that, as j is proper, convex and l.s.c., the same properties hold true for J. Then setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M45">View MathML</a>

(2.4)

it is clear that ψ is proper, convex and l.s.c. on X.

Further, we define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M46">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M47">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M48">View MathML</a>

(2.5)

It is a simple matter to check that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M49">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M50">View MathML</a>

(2.6)

The energy functional associated to problem (1.1), (1.2) is

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M51">View MathML</a>

with ψ in (2.4) and Φ given by (2.5).

Proposition 2.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M52">View MathML</a>is a critical point of the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M53">View MathML</a>in the sense that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M54">View MathML</a>

(2.7)

thenxis a solution of problem (1.1), (1.2).

Proof In (2.7), we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M56">View MathML</a>; then dividing by s and letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M57">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M58">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M59">View MathML</a> is the directional derivative of the convex function J at x in the direction of w. By virtue of (2.3), the above inequality becomes

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M60">View MathML</a>

Using (2.2), (2.6) and the summation by parts formula, a straightforward computation shows that

(2.8)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M62">View MathML</a>. Thus, we infer

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M63">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M62">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M65">View MathML</a>. This implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M66">View MathML</a>

(2.9)

To prove that x satisfies condition (1.2), we multiply the equality (2.9) by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M67">View MathML</a>. Then summing from 1 to T and using (2.8), one obtains

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M68">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M62">View MathML</a>. Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M62">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M71">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M72">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M73">View MathML</a> are arbitrarily chosen, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M74">View MathML</a>

which, by a standard result from convex analysis, means that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M75">View MathML</a>

and the proof is complete. □

From now on, we will use the following notations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M76">View MathML</a>

Remark 2.2 It is easy to check that for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M52">View MathML</a> and any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M37">View MathML</a>, we have

(2.10)

(2.11)

3 Ground state solutions

We begin by a result which states that the energy functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M53">View MathML</a> has a minimum point in X provided that the potential of the nonlinearity f lies asymptotically on the left of the first eigenvalue like constant

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M82">View MathML</a>

(3.1)

Theorem 3.1If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M83">View MathML</a>

(3.2)

then problem (1.1), (1.2) has at least one solution which minimizes<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M53">View MathML</a>onX.

Proof By the continuity of Φ and the lower semicontinuity of ψ, we have that the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M53">View MathML</a> is sequentially l.s.c. on X. It remains to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M53">View MathML</a> is coercive on X. Then, by the direct method in calculus of variations, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M53">View MathML</a> is bounded from below and attains its infimum at some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M52">View MathML</a>, which, by virtue of ([20], Proposition 1.1) and Proposition 2.1, is a solution of problem (1.1), (1.2).

From (3.2) there are constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M89">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M90">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M91">View MathML</a>

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M92">View MathML</a>, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M93">View MathML</a>. On the other hand, by the continuity of F, there is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M94">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M95">View MathML</a>

Hence, we infer

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M96">View MathML</a>

To prove the coercivity of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M53">View MathML</a>, from the above inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M98">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M99">View MathML</a>.

If <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M100">View MathML</a>, using (2.11) from Remark 2.2, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M101">View MathML</a>

(3.3)

In the case <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M92">View MathML</a>, by virtue of (2.11) and (3.1), for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M103">View MathML</a>, one obtains

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M104">View MathML</a>

(3.4)

which, using again (3.1), implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M105">View MathML</a>

(3.5)

In both cases, by virtue of (3.3) and (3.5), there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M106">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M107">View MathML</a>

On the other hand, as j is convex and l.s.c., it is bounded from below by an affine functional. Therefore, on account of (2.3), there are positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M108">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M109">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M110">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M111">View MathML</a>

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M112">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M113">View MathML</a>. Since any norm on X is equivalent to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M114">View MathML</a>, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M115">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M116">View MathML</a>

Consequently,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M117">View MathML</a>

meaning that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M53">View MathML</a> is coercive on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M119">View MathML</a> and the proof is complete. □

In order to give an application of Theorem 3.1, we consider the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M120">View MathML</a>

(3.6)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M121">View MathML</a> is a continuous function and λ is a positive parameter.

Corollary 3.2Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M92">View MathML</a>andgsatisfies the growth condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M123">View MathML</a>

(3.7)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M124">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M125">View MathML</a>are constants and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M126">View MathML</a>. The following hold true:

(i) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M127">View MathML</a>, then problem (3.6) has a solution for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M128">View MathML</a>;

(ii) if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M129">View MathML</a>, then there is some<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M130">View MathML</a>such that for any<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M131">View MathML</a>, problem (3.6) has a solution.

Proof We apply Theorem 3.1 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M132">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M133">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M134">View MathML</a>. From (3.7), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M135">View MathML</a>

Thus, we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M136">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M133">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M134">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M139">View MathML</a>. So, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M127">View MathML</a>, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M141">View MathML</a>

Also, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M129">View MathML</a>, setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M143">View MathML</a>

(3.8)

it is easy to see that condition (3.2) is fulfilled for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M131">View MathML</a>. □

Remark 3.3

(i) Note that a valid <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M145">View MathML</a> in Corollary 3.2(ii) is given by formula (3.8).

(ii) Theorem 5 in [11] is an immediate consequence of Corollary 3.2 with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M28">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M29">View MathML</a>.

4 Mountain pass type solutions

In this section, we deal with the existence of nontrivial solutions for the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M148">View MathML</a>

(4.1)

associated with the potential boundary condition (1.2). Here, f and j are as in the case of the previous problem (1.1), (1.2) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M149">View MathML</a> is a given function. The main tool in obtaining such a result will be the Mountain Pass Theorem [20].

To treat problem (4.1), (1.2), instead of φ, there will be <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M150">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M151">View MathML</a>

(4.2)

which is convex, of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M41">View MathML</a> on X, and its derivative is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M153">View MathML</a>

(4.3)

Then, setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M154">View MathML</a>, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M155">View MathML</a>

(4.4)

with J given by (2.3) and Φ in (2.5).

By means of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M156">View MathML</a> in (3.1), we define the constants

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M157">View MathML</a>

(4.5)

Lemma 4.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M158">View MathML</a>and there exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M159">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M160">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M161">View MathML</a>

(4.6)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M162">View MathML</a>

(4.7)

then the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M163">View MathML</a>defined in (4.4) satisfies the Palais-Smale condition ((PS) condition for short) on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M164">View MathML</a>, i.e., every sequence<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M165">View MathML</a>for which<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M166">View MathML</a>and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M167">View MathML</a>

(4.8)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M168">View MathML</a>, possesses a convergent subsequence.

Proof Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M165">View MathML</a> be a sequence for which <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M166">View MathML</a> and (4.8) holds true with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M168">View MathML</a>. Since X is finite dimensional, it is sufficient to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M172">View MathML</a> is bounded. In order to show this, we may assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M173">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M174">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M175">View MathML</a>. By virtue of (2.11), (3.1) and (4.5), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M176">View MathML</a>

(4.9)

From (2.3) and (4.6), it follows

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M177">View MathML</a>

(4.10)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M178">View MathML</a>. Using (4.7) we deduce that, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M175">View MathML</a>, it holds

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M180">View MathML</a>

(4.11)

Clearly, there is a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M181">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M182">View MathML</a>

(4.12)

Further, setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M183">View MathML</a> in (4.8), dividing by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M56">View MathML</a> and then letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M57">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M186">View MathML</a>

(4.13)

Using (4.12) and (4.13), we deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M187">View MathML</a>

and by virtue of (4.10), (4.11), (4.3) and (4.9), we have

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M159">View MathML</a>, we infer that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M172">View MathML</a> is bounded and the proof is complete. □

Now, we can state the following result of Ambrosetti-Rabinowitz type [21].

Theorem 4.2Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M158">View MathML</a>and, in addition,

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M192">View MathML</a>;

(ii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M193">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M194">View MathML</a>;

(iii) there are constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M159">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M160">View MathML</a>such that (4.6) holds true and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M197">View MathML</a>

(4.14)

Then, problem (4.1), (1.2) has a nontrivial solution.

Proof Without loss of generality, we may assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M198">View MathML</a>

(4.15)

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M199">View MathML</a>. From (i), (2.3) and (4.15), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M200">View MathML</a>

(4.16)

From Lemma 4.1 and (iii), the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M163">View MathML</a> satisfies the (PS) condition on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M202">View MathML</a>.

Next, we shall prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M163">View MathML</a> has a ‘mountain pass’ geometry:

(a) there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M204">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M205">View MathML</a> if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M206">View MathML</a>;

(b) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M207">View MathML</a> for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M208">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M209">View MathML</a>.

By the equivalence of the norms on X, there is some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M210">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M211">View MathML</a>

(4.17)

Using (ii) we can find constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M212">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M213">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M214">View MathML</a>

(4.18)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M52">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M216">View MathML</a>, be arbitrarily chosen. From (4.17) and (4.18), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M217">View MathML</a>

which implies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M218">View MathML</a>

Now, using (2.10) and (3.1), we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M219">View MathML</a>

(4.19)

By virtue of (3.1), (4.5), (4.2) and (4.19), we deduce

On account of (4.16), we infer that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M221">View MathML</a>

(4.20)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M222">View MathML</a>, and condition (a) is fulfilled.

Our next task is to prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M163">View MathML</a> satisfies condition (b). To this end, let us first observe that, by virtue of (4.14), there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M224">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M225">View MathML</a>

(4.21)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M226">View MathML</a> be such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M227">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M228">View MathML</a>. Using (2.11) and (4.5), one obtains

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M229">View MathML</a>

(4.22)

From (4.15), we have that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M230">View MathML</a>

which, together with (4.21) and (4.22) for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M231">View MathML</a>, gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M232">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M233">View MathML</a> because <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M159">View MathML</a>. Hence, we can choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M235">View MathML</a> large enough to satisfy <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M236">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M237">View MathML</a>, with μ entering in (4.20). This means that condition (b) is satisfied with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M238">View MathML</a>. □

5 An application

In this section, we show how Theorem 4.2 can be applied to derive the existence of nontrivial solutions for equation (4.1) associated with some concrete boundary conditions.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M239">View MathML</a> be a convex and Gâteaux differentiable function with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M240">View MathML</a>, where dg denotes the differential of g. Also, given a nonempty closed convex cone <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M241">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M242">View MathML</a> the normal cone to K at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M243">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M244">View MathML</a>

The equation (4.1) is considered to be associated with the boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M245">View MathML</a>

(5.1)

We set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M246">View MathML</a>

Theorem 5.1If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M247">View MathML</a>is continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M248','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M248">View MathML</a>and, in addition, we assume that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M249','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M249">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M194">View MathML</a>;

(ii) there are constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M159">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M160">View MathML</a>such that (4.14) holds true and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M253">View MathML</a>

(5.2)

then problem (4.1), (5.1) has a nontrivial solution.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M254">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M243">View MathML</a>, Theorem 4.2 applies with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M256">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M257">View MathML</a>. □

Remark 5.2 Conditions (5.1) allow various possible choices of g and K, which, among others, recover classical boundary conditions. For instance, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M258">View MathML</a>, then the homogeneous boundary conditions

are obtained by choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M29">View MathML</a>, respectively <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M30">View MathML</a>. If, in addition, p is T-periodic, then taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M263">View MathML</a>, we get

respectively. If the T-periodicity condition is not assumed, then we only have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M265">View MathML</a>

instead of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M266','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M266">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M267">View MathML</a>, respectively. As <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M258">View MathML</a>, in these four cases, condition (5.2) is automatically satisfied with any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M269">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M270">View MathML</a>.

Also, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M271','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M271">View MathML</a> are given, then with g defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M272">View MathML</a>

and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M30">View MathML</a>, we deduce the Sturm-Liouville type boundary conditions

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M274">View MathML</a>

In this case, (5.2) is fulfilled with any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M275">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M270">View MathML</a>.

Therefore, sufficient conditions ensuring the existence of nontrivial solutions of (4.1) subjected to one of the above boundary conditions can be easily stated by means of Theorem 5.1.

Remark 5.3 It is worth pointing out that in the cases of Dirichlet and antiperiodic boundary conditions, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M277">View MathML</a> is allowed to be =0, and hence, r may be ≥0 on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M278">View MathML</a>; while in the Neumann, periodic and Sturm-Liouville cases, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M277','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M277">View MathML</a> must be >0, meaning <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M280','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M280">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/104/mathml/M278">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.

Acknowledgements

Dedicated to Professor Jean Mawhin for his 70th anniversary.

The research of CŞ was supported by the strategic grant POSDRU/CPP107/DMI1.5/S/78421, Project ID 78421 (2010), co-financed by the European Social Fund - Investing in People, within the Sectoral Operational Programme Human Resources Development 2007-2013. Also, the support for CB and PJ from the grant TE-PN-II-RU-TE-2011-3-0157 (CNCS-Romania) is gratefully acknowledged.

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