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Weak solutions for the singular potential wave system

Tacksun Jung1 and Q-Heung Choi2*

Author Affiliations

1 Department of Mathematics, Kunsan National University, Kunsan, 573-701, Korea

2 Department of Mathematics Education, Inha University, Incheon, 402-751, Korea

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Boundary Value Problems 2014, 2014:7  doi:10.1186/1687-2770-2014-7


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/7


Received:13 February 2013
Accepted:9 December 2013
Published:7 January 2014

© 2014 Jung and Choi; 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

We investigate the existence of weak solutions for a class of the system of wave equations with singular potential nonlinearity. We obtain a theorem which shows the existence of nontrivial weak solution for a class of the wave system with singular potential nonlinearity and the Dirichlet boundary condition. We obtain this result by using the variational method and critical point theory for indefinite functional.

MSC: 35L51, 35L70.

Keywords:
class of wave system; singular potential nonlinearity; Dirichlet boundary condition; variational method; critical point theorem for indefinite functional; <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M1">View MathML</a> condition

1 Introduction

Let D be an open subset in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M2">View MathML</a> with compact complement <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M4">View MathML</a>. In this paper we investigate the multiplicity of the solutions for a class of the system of nonlinear wave equations with the Dirichlet boundary condition and periodic condition:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M5">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M6">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M7">View MathML</a>. We assume that G satisfies the following conditions:

(G1) There exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M8">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M9">View MathML</a>

(G2) There is a neighborhood Z of C in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M2">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M11">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M12">View MathML</a> is the distance function from U to C and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M13">View MathML</a> is a constant. The system (1.1) can be rewritten as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M14">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M15">View MathML</a>.

Remark We have a simple example satisfying the above conditions (G1)-(G2):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M16">View MathML</a>

Our main result is the following.

Theorem 1.1Assume that the nonlinear termGsatisfies conditions (G1)-(G2). Then system (1.1) has at least one nontrivial weak solution.

For the proof of Theorem 1.1, we approach the variational method and use the critical point theory for indefinite functional. In Section 2, we introduce a Banach space and the associated functional I of (1.1), and recall the critical point theory for indefinite functional. In Section 3, we prove that I satisfies the geometric assumptions of the critical point theorem for indefinite functional and prove Theorem 1.1.

2 Variational approach

The eigenvalue problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M17">View MathML</a>

(2.1)

has infinitely many eigenvalues

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M18">View MathML</a>

and corresponding normalized eigenfunctions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M20">View MathML</a>, given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M21">View MathML</a>

Let Ω be the square <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M22">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M23">View MathML</a> be the Hilbert space defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M24">View MathML</a>

The set of functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M25">View MathML</a> is an orthonormal basis in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M23">View MathML</a>. Let us denote an element v, in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M23">View MathML</a>, as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M28">View MathML</a>

and we define a subspace E of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M23">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M30">View MathML</a>

This is a complete normed space with a norm

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M31">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M32">View MathML</a> is unbounded from above and from below and has no finite accumulation point, it is convenient for the following to rearrange the eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M33">View MathML</a> by increasing magnitude: from now on we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M34">View MathML</a> the sequence of negative eigenvalues of (2.1), by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M35">View MathML</a> the sequence of positive ones, so that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M36">View MathML</a>

We will denote by the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M37">View MathML</a> all the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M38">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M39">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M40">View MathML</a> be an orthonormal system of the eigenfunctions associated with the eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M41">View MathML</a>. We will denote by the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M42">View MathML</a> the sequences <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M43">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M44">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M45">View MathML</a> be the span of closure of eigenfunctions associated with positive eigenvalues and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M46">View MathML</a> be the span of closure of eigenfunctions associated with negative eigenvalues. Let H be the n Cartesian product space of E, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M47">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M48">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M49">View MathML</a> be the subspaces on which the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M50">View MathML</a>

is positive definite and negative definite, respectively. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M51">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M52">View MathML</a> be the projection from H onto <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M48">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M54">View MathML</a> be the projection from H onto <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M49">View MathML</a>. The norm in H is given by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M56">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M57">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M7">View MathML</a>.

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M60">View MathML</a> be a sequence of closed finite dimensional subspace of H with the following assumptions: <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M61">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M63">View MathML</a> for all n (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M65">View MathML</a> are subspaces of H), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M68">View MathML</a> is dense in H.

In this paper we are trying to find the weak solutions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M69">View MathML</a> of system (1.1), that is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M70">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M71">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M72">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M73">View MathML</a>

Let us introduce an open set of the Hilbert space H as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M74">View MathML</a>

Let us consider the functional on X

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M75">View MathML</a>

(2.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M76">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M77">View MathML</a>. The Euler equation for (2.1) is (1.1). By the following Lemma 2.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M78">View MathML</a>, and so the weak solutions of system (1.1) coincide with the critical points of the associated functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M79">View MathML</a>.

Lemma 2.1Assume thatGsatisfies conditions (G1)-(G2). Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M79">View MathML</a>is continuous and Fréchet differentiable inXwith Fréchet derivative

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M81">View MathML</a>

Moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M82">View MathML</a>. That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M83">View MathML</a>.

Proof First we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M79">View MathML</a> is continuous. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M85">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M86">View MathML</a>

We have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M87">View MathML</a>

(2.3)

Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M88">View MathML</a>

Next we shall prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M79">View MathML</a> is Fréchet differentiable in X. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M85">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M91">View MathML</a>

Thus by (2.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M92">View MathML</a>

(2.4)

Similarly, it is easily checked that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M83">View MathML</a>. □

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M94">View MathML</a>

Lemma 2.2Assume thatGsatisfies conditions (G1)-(G2). Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M95">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M96">View MathML</a>weakly inXwith<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M97">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M98">View MathML</a>.

Proof To prove the conclusion, it suffices to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M99">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M100">View MathML</a> is bounded from below, it suffices to prove that there is a subset <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M101">View MathML</a> of Ω such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M102">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M97">View MathML</a> means that there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M104">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M105">View MathML</a>. Let us set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M106">View MathML</a>

By (G1) and (G2), there exists a constant B such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M107">View MathML</a>

Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M108">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M109">View MathML</a>. By Schwarz’s inequality, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M110">View MathML</a>

Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M111">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M112">View MathML</a>

Since the embedding <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M113">View MathML</a> is compact, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M114">View MathML</a>

Thus by Fatou’s lemma, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M115">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M116">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M117">View MathML</a>

so we prove the lemma. □

We recall the critical point theorem for the indefinite functional (cf.[1]).

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M118">View MathML</a>

Theorem 2.1 (Critical point theorem for the indefinite functional)

LetXbe a real Hilbert space with<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M119">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M120">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M78">View MathML</a>satisfies (PS), and

(I1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M122">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M123">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M124">View MathML</a>is bounded and self-adjoint, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M125">View MathML</a>,

(I2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M126">View MathML</a>is compact, and

(I3) there exists a subspace<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M127">View MathML</a>and sets<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M129">View MathML</a>and constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M130">View MathML</a>such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M131">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M132">View MathML</a>,

(ii) Qis bounded and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M133">View MathML</a>,

(iii) Sand∂Qlink.

ThenIpossesses a critical value<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M134">View MathML</a>.

3 Proof of Theorem 1.1

We shall show that the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M79">View MathML</a> satisfies the geometric assumptions of the critical point theorem for indefinite functional.

Lemma 3.1 (Palais-Smale condition)

Assume thatGsatisfies conditions (G1) and (G2). Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M136">View MathML</a>satisfies the (PS) condition inX.

Proof We shall prove the lemma by contradiction. We suppose that there exists a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M137">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M138">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M139">View MathML</a>

(3.1)

or equivalently

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M140">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M141">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M142">View MathML</a> is a compact operator. We claim that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M143">View MathML</a>, up to a subsequence, converges. It suffices to prove that the sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M143">View MathML</a> is bounded in X. By contradiction, we suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M145">View MathML</a>. Then, for large k, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M146">View MathML</a>

(3.2)

It follows from (3.2) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M147">View MathML</a>

(3.3)

Let us set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M148">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M149">View MathML</a>, and hence the subsequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M150">View MathML</a>, up to a subsequence, converges weakly to W with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M151">View MathML</a>. By (3.1), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M152">View MathML</a>

(3.4)

Letting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M153">View MathML</a> in (3.4), by (3.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M154">View MathML</a>

(3.5)

Thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M155">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M156">View MathML</a>

and by (3.5), W is the weak solution of the equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M157">View MathML</a>

We claim that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M158">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M159">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M160">View MathML</a> is the kernel of A. In fact, let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M161">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M162">View MathML</a>. Then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M163">View MathML</a>

We note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M164">View MathML</a>

Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M158">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M166">View MathML</a>, which is absurd to the fact that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M151">View MathML</a>. Thus <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M143">View MathML</a> is bounded. Thus the subsequence, up to a subsequence, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M169">View MathML</a> converges weakly to U in X. By Lemma 2.2, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M170">View MathML</a> and that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M171">View MathML</a> is bounded. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M142">View MathML</a> is compact and (3.1) holds, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M143">View MathML</a> converges strongly to U. Thus we prove the lemma. □

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M174">View MathML</a>

Lemma 3.2Assume thatGsatisfies conditions (G1) and (G2). Then there exist sets<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M175">View MathML</a>with radius<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M176">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M177">View MathML</a>and constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M178">View MathML</a>such that

(i) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M175">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M180">View MathML</a>,

(ii) Qis bounded and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M181">View MathML</a>,

(iii) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M182">View MathML</a>and∂Qlink.

Proof (i) Let us choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M183">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M184">View MathML</a>. By (G1), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M185','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M185">View MathML</a> is bounded above and there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M186">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M187">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M186">View MathML</a>. Then there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M176">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M178">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M191">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M192">View MathML</a>.

(ii) Let us choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M193">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M194','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M194">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M195">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M196">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M197">View MathML</a>. We note that:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M198','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M198">View MathML</a>

By (G2), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M199">View MathML</a> is bounded from below. Thus by Lemma 2.2, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M13">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M201">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M202">View MathML</a>

We can choose a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M203">View MathML</a> such that if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M204">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M205">View MathML</a>. Thus we prove the lemma. □

Proof of Theorem 1.1 By Lemma 2.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M79">View MathML</a> is continuous and Fréchet differentiable in X and, moreover, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M82">View MathML</a>. By Lemma 2.2, if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M95">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M96">View MathML</a> weakly in X with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M97">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M98">View MathML</a>. By Lemma 3.1, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M136">View MathML</a> satisfies the (PS) condition. By Lemma 3.2, there exist sets <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M175">View MathML</a> with radius <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M176">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M177">View MathML</a> and constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M178">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M217">View MathML</a>, Q is bounded and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M181">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M182">View MathML</a> and ∂Q link. By the critical point theorem, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M79">View MathML</a> possesses a critical value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/7/mathml/M221">View MathML</a>. Thus (1.1) has at least one nontrivial weak solution. Thus we prove Theorem 1.1 □

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, checked and approved the final manuscript.

Acknowledgements

This work (Tacksun Jung) was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (KRF-2010-0023985).

References

  1. Benci, V, Rabinowitz, PH: Critical point theorems for indefinite functionals. Invent. Math.. 52, 241–273 (1979). Publisher Full Text OpenURL