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Multiplicity results for nonlinear mixed boundary value problem

Giuseppina D’Aguì

Author affiliations

Department of Civil, Computer, Construction, Environmental Engineering and Applied Mathematics, University of Messina, Messina, 98166, Italy

Citation and License

Boundary Value Problems 2012, 2012:134  doi:10.1186/1687-2770-2012-134


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/134


Received:20 July 2012
Accepted:26 October 2012
Published:14 November 2012

© 2012 D’Aguì; licensee Springer

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

Abstract

The aim of this paper is to establish multiplicity results of nontrivial and nonnegative solutions for mixed boundary value problems with the Sturm-Liouville equation. The approach is based on variational methods.

MSC: 34B15.

Keywords:
boundary value problem; mixed conditions

1 Introduction

The aim of this paper is to establish existence results of two and three nontrivial solutions for Sturm-Liouville problems with mixed conditions involving the ordinary p-Laplacian. We consider the following problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M1">View MathML</a>

(P)

with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M3">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M4">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M5">View MathML</a>. Here the nonlinearity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M6">View MathML</a> is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M7">View MathML</a>-Carathéodory function and λ is a real positive parameter.

The existence of at least one solution for problem (P) has been obtained in [1], where only a unique algebraic condition on the nonlinear term is assumed (see [[1], Theorem 1.1]). In the present paper, first we obtain the existence of two solutions by combining an algebraic condition on f of type contained in [1] with the classical Ambrosetti-Rabinowitz condition.

(AR): There exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M9">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M10">View MathML</a>

The role of (AR) is to ensure the boundness of the Palais-Smale sequences for the Euler-Lagrange functional associated to the problem. This is very crucial in the applications of critical point theory. Subsequently, an existence result of three positive solutions is obtained combining two algebraic conditions which guarantee the existence of two local minima for the Euler-Lagrange functional and applying the mountain pass theorem as given by Pucci and Serrin (see [2]) to ensure the existence of the third critical point.

Many mathematical models give rise to problems for which only nonnegative solutions make sense; therefore, many research articles on the theory of positive solutions have appeared. For a complete overview on this subject, we refer to the monograph [3].

In this paper, we also present, as a consequence of our main theorems, some results on the existence of nonnegative solutions for a particular problem of type

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M11">View MathML</a>

(P1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M12">View MathML</a> is such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M13">View MathML</a> a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M15">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M16">View MathML</a> is a nonnegative continuous function. In particular, we obtain for such a problem the existence of at least three nonnegative solutions by requiring that the function g has a superlinear behavior at zero, a sublinear behavior at infinity, and a particular growth in a suitable interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M17">View MathML</a>. By a similar approach, in [4], the authors obtain the existence of multiple solutions for a Neumann elliptic problem.

Multiplicity results for a mixed boundary value problem have been studied by several authors (see, for instance, [5-8] and references therein). In [5], the authors establish multiplicity results for problem (P), when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M18">View MathML</a>, and, in particular, they obtain the existence of three solutions, one of which can be trivial. On the contrary, our results (Theorems 3.5 and 3.6) guarantee the existence of three nonnegative and nontrivial solutions.

In [7], by using a fixed point theorem, the existence of at least three solutions for a mixed boundary problem with the equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M19">View MathML</a> is obtained, by requiring, among other things, the boundness of f in a right neighborhood of zero (hypothesis (H6), Theorem 3.1), instead in our results (Theorems 3.5 and 3.6) the nonlinearity can blow up at zero.

Here, as an example, we present the following result which is a particular case of Theorem 3.6.

Theorem 1.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M20">View MathML</a>be a nonnegative continuous function such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M21">View MathML</a>

(1.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M22">View MathML</a>

(1.2)

Then, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M23">View MathML</a>, the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M24">View MathML</a>

admits at least three classical nonnegative and nontrivial solutions.

2 Preliminaries and basic notations

Our main tools are Theorems 2.1 and 2.2, consequences of the existence result of a local minimum [[9], Theorem 3.1] which is inspired by the Ricceri variational principle (see [10]). For more information on this topic see, for instance, [11] and [12].

Given a set X and two functionals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M25">View MathML</a>, put

(2.1)

(2.2)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M28">View MathML</a>, with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M29">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M30">View MathML</a>

(2.3)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M31">View MathML</a>.

Theorem 2.1 [[9], Theorem 5.1]

LetXbe a reflexive real Banach space; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M32">View MathML</a>be a sequentially weakly lower semicontinuous, coercive, and continuously Gâteaux differentiable function whose Gâteaux derivative admits a continuous inverse on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M33">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M34">View MathML</a>be a continuously Gâteaux differentiable function whose Gâteaux derivative is compact. Put<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M35">View MathML</a>and assume that there are<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M37">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M29">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M39">View MathML</a>

(2.4)

whereβand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M40">View MathML</a>are given by (2.1) and (2.2).

Then, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M41">View MathML</a>, there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M42">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M43">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M44">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M45">View MathML</a>.

Theorem 2.2 [[9], Theorem 5.3]

LetXbe a real Banach space; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M46">View MathML</a>be a continuously Gâteaux differentiable function whose Gâteaux derivative admits a continuous inverse on<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M33">View MathML</a>; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M34">View MathML</a>be a continuously Gâteaux differentiable function whose Gâteaux derivative is compact. Fix<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M49">View MathML</a>and assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M50">View MathML</a>

(2.5)

whereρis given by (2.3), and for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M51">View MathML</a>, the function<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M52">View MathML</a>is coercive.

Then, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M53">View MathML</a>, there is<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M54">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M55">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M56">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M57">View MathML</a>.

Now, consider problem (P) and assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M58">View MathML</a> with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M59">View MathML</a>

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M60">View MathML</a> endowed with the norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M61">View MathML</a>

Throughout the sequel, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M62">View MathML</a> is an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M7">View MathML</a>-Carathéodory function. We recall that a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M64">View MathML</a> is said to be an <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M7">View MathML</a>-Carathéodory function if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M66">View MathML</a> is measurable for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M68">View MathML</a> is continuous for almost every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M69">View MathML</a>, and for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M70">View MathML</a>, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M71">View MathML</a>. Clearly, if f is continuous in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M72">View MathML</a>, then it is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M7">View MathML</a>-Carathéodory.

Put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M74">View MathML</a>

Moreover, it is well known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M75">View MathML</a> is compactly embedded in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M76">View MathML</a> and one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M77">View MathML</a>

(2.6)

We use the following notations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M78">View MathML</a>

In order to study problem (P), we introduce the functionals <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M79">View MathML</a> defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M80">View MathML</a>

(2.7)

Clearly, the critical points of the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M81">View MathML</a> on X are weak solutions of problem (P). We recall that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M82">View MathML</a> is a weak solution of problem (P) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M83">View MathML</a> satisfies the following condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M84">View MathML</a>

Clearly, if f is continuous, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M85">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M86">View MathML</a>, the weak solutions for (P) are classical solutions.

3 Main results

In this section we present our main results.

Given two nonnegative constants c, d such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M87">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M88">View MathML</a>

put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M89">View MathML</a>

Theorem 3.1Under the following conditions:

(i) there exist three constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M91">View MathML</a>, d, with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M92">View MathML</a>

(3.1)

such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M93">View MathML</a>

and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M94">View MathML</a><a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M95">View MathML</a>;

(ii) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M8">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M9">View MathML</a>such that

For each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M99">View MathML</a>, problem (P) admits at least two nontrivial weak solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M100">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M101">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M100">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M103">View MathML</a>.

Proof The proof of this theorem is divided into two steps. In the first part, by applying Theorem 2.1, we prove the existence of a local minimum for the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M81">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M105">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M106">View MathML</a> are functionals given in (2.7) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M83">View MathML</a>. Obviously, Φ and Ψ satisfy all regularity assumptions requested in Theorem 2.1, and the critical points in X of the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M108">View MathML</a> are exactly the weak solutions of problem (P). To this end, we verify condition (2.4) of Theorem 2.1.

Define the following function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M109">View MathML</a>, by setting

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M110">View MathML</a>

and estimate <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M111">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M112">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M113">View MathML</a>

(3.2)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M114">View MathML</a>

(3.3)

Fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M90">View MathML</a>, d, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M91">View MathML</a> satisfying (3.1) and put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M117">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M118">View MathML</a>.

From (3.1), one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M119">View MathML</a>.

Moreover, for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M83">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M121">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M122">View MathML</a>

(3.4)

Hence,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M123">View MathML</a>

(3.5)

Now, arguing as before, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M124">View MathML</a>

(3.6)

From hypothesis (i) and bearing in mind (3.3), (3.2), (3.5), and (3.6), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M125">View MathML</a>

From Theorem 2.1, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M126">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M127">View MathML</a> admits at least one critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M100">View MathML</a> which is a local minimum such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M129">View MathML</a>

Now, we prove the existence of the second local minimum distinct from the first one. To this end, we must show that the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M130">View MathML</a> satisfies the hypotheses of the mountain pass theorem.

Clearly, the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M81">View MathML</a> is of class <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M132">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M133">View MathML</a>.

From the first part of the proof, we can assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M100">View MathML</a> is a strict local minimum for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M108">View MathML</a> in X. Therefore, there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M136">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M137">View MathML</a>, so condition [[13], (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M138">View MathML</a>), Theorem 2.2] is verified.

Now, choosing any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M139">View MathML</a>, from (ii) one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M140">View MathML</a>

as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M141">View MathML</a>, so condition [[13], (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M142">View MathML</a>), Theorem 2.2] is verified. Moreover, by standard computations, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M108">View MathML</a> satisfies the Palais-Smale condition. Hence, the classical theorem of Ambrosetti and Rabinowitz ensures a critical point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M101">View MathML</a> of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M145">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M146">View MathML</a>. So, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M100">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M101">View MathML</a> are two distinct weak solutions of (P) and the proof is complete. □

Remark 3.1 We observe that in literature the existence of at least one nontrivial solution for differential problems is obtained associating to the classical Ambrosetti-Rabinowitz condition a hypothesis on the nonlinear term of type <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M149">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M150">View MathML</a>. This implies that the problem possesses also the trivial solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M151">View MathML</a>. In Theorem 3.1, we find a nontrivial solution of the problem that actually is a proper local minimum of the Euler-Lagrange functional associated to the problem different from zero.

Now, we present an application of Theorem 2.2 which we will use to obtain multiple solutions.

Theorem 3.2Assume that there exist two constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M153">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M154">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M155">View MathML</a>

(3.7)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M156">View MathML</a>

(3.8)

Then, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M157">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M158">View MathML</a>

problem (P) admits at least one nontrivial weak solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M159">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M160">View MathML</a>.

Proof The functionals Φ and Ψ satisfy all regularity assumptions requested in Theorem 2.2. Moreover, by standard computations, condition (3.8) implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M108">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M162">View MathML</a>, is coercive. So, our aim is to verify condition (2.5) of Theorem 2.2. To this end, put

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M163">View MathML</a>

Arguing as in the proof of Theorem 3.1, we obtain that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M164">View MathML</a>

So, from our assumption, it follows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M165">View MathML</a>.

Hence, from Theorem 2.2 for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M166">View MathML</a>, the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M108">View MathML</a> admits at least one local minimum <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M159">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M160">View MathML</a> and our conclusion is achieved. □

Remark 3.2 We point out that the same statement of above given result can be obtained by using a classical direct methods theorem (see [14]), but in addition we get the location of the solution, hence in particular the solution is nontrivial.

Now, we point out some results when the nonlinear term is with separable variables. To be precise, let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M12">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M13">View MathML</a> a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M14">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M15">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M16">View MathML</a> be a nonnegative continuous function,

consider the following boundary value problem:

We observe that the following results give the existence of multiple nonnegative solutions since the nonlinear term is supposed to be nonnegative. In order to justify what has been said above, we point out the following weak maximum principle.

Lemma 3.1Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M176">View MathML</a>is a weak solution of problem (P1), then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M177">View MathML</a>is nonnegative.

Proof We claim that a weak solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M177">View MathML</a> is nonnegative. In fact, arguing by a contradiction and setting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M179">View MathML</a>, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M180">View MathML</a>. Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M181">View MathML</a>, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M182">View MathML</a> (see, for instance, [[15], Lemma 7.6]). So, taking into account that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M183">View MathML</a> is a weak solution and by choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M184">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M185">View MathML</a>

that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M186">View MathML</a> which is absurd. Hence, our claim is proved. □

Corollary 3.1Assume that

(i′) there exist three nonnegative constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M90">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M91">View MathML</a>, d, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M189">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M190">View MathML</a>

(3.9)

(ii′) there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M191">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M192">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M193">View MathML</a>

Then, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M194">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M195">View MathML</a>

problem (P1) admits at least two nonnegative weak solutions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M100">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M101">View MathML</a>such that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M198">View MathML</a>.

Theorem 3.3Assume that there exist two positive constantsc, d, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M199">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M200">View MathML</a>

(3.10)

Further, suppose that there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M191">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M192">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M203">View MathML</a>

Then, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M204">View MathML</a>, problem (P1) admits at least two nonnegative weak solutions.

Proof Our aim is to apply Corollary 3.1. To this end, we pick <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M205">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M206">View MathML</a>. From (3.10), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M207">View MathML</a>

On the other hand, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M208">View MathML</a>

Hence, from Corollary 3.1 and taking (2.6) into account, the conclusion follows. □

A further consequence of Theorem 3.1 is the following result.

Theorem 3.4Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M209">View MathML</a>

(3.11)

and there are constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M210">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M192">View MathML</a>such that, for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M212">View MathML</a>, one has

Then, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M214">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M215">View MathML</a>, problem (P1) admits at least two nonnegative weak solutions.

Proof Fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M216">View MathML</a>. Then there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M217">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M218">View MathML</a>. From (3.11), there is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M219">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M220">View MathML</a>. Hence, Theorem 3.3 ensures the conclusion. □

Next, as a consequence of Theorems 3.3 and 3.2, the following theorem of the existence of three solutions is obtained.

Theorem 3.5Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M221">View MathML</a>

(3.12)

Moreover, assume that there exist four positive constantsc, d, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M153">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M224">View MathML</a>, such that (3.10),

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M225">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M226">View MathML</a>

(3.13)

are satisfied.

Then, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M227">View MathML</a>, problem (P1) admits at least three weak nonnegative solutions.

Proof First, we observe that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M228">View MathML</a> owing to (3.13). Next, fix <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M194">View MathML</a>. Theorem 3.3 ensures a nontrivial weak solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M177">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M231">View MathML</a> which is a local minimum for the associated functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M108">View MathML</a>, as well as Theorem 3.2 guarantees a nontrivial weak solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M159">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M234','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M234">View MathML</a> which is a local minimum for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M108">View MathML</a>. Hence, the mountain pass theorem as given by Pucci and Serrin (see [2]) ensures the conclusion. □

Theorem 3.6Assume that

(3.14)

(3.15)

Further, assume that there exist two positive constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M238">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M153">View MathML</a>, with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M240','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M240">View MathML</a>, such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M241">View MathML</a>

(3.16)

Then, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M242">View MathML</a>, problem (P1) admits at least three weak nonnegative solutions.

Proof Clearly, (3.15) implies (3.8). Moreover, by choosing d small enough and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M243">View MathML</a>, simple computations show that (3.14) implies (3.10). Finally, from (3.16) we get (3.7) and also (3.13). Hence, Theorem 3.5 ensures the conclusion. □

Finally, we present two examples of problems that admit multiple solutions owing to Theorems 3.4 and 3.6.

Example 3.1 Owing to Theorem 3.4, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M244','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M244">View MathML</a>, the problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M245','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M245">View MathML</a>

admits at least two nonnegative solutions. In fact, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M246">View MathML</a> and (AR) is satisfied as a simple computation shows. Moreover, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M247','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M247">View MathML</a>.

Example 3.2 Consider the following problem:

It has three nonnegative solutions. In fact, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M20">View MathML</a> be a function defined as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M250','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M250">View MathML</a>

Owing to Theorem 3.6, the following problem

admits three nonnegative classical solutions. In fact, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M252','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M252">View MathML</a>

Moreover, taking into account that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M253">View MathML</a>, by choosing <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M254">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M255">View MathML</a>, one has <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M256">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M257','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M257">View MathML</a>.

So, it is clear that any nonnegative solution u of problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M258">View MathML</a>) is also a solution of problem (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M259','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/134/mathml/M259">View MathML</a>).

Competing interests

The author declares that she has no competing interests.

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