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Infinitely many periodic solutions for subquadratic second-order Hamiltonian systems

Hua Gu* and Tianqing An

Author Affiliations

College of Science, Hohai University, Nanjing, 210098, China

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Boundary Value Problems 2013, 2013:16  doi:10.1186/1687-2770-2013-16

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/16


Received:8 November 2012
Accepted:10 January 2013
Published:6 February 2013

© 2013 Gu and An; licensee Springer

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

Abstract

In this paper, we investigate the existence of infinitely many periodic solutions for a class of subquadratic nonautonomous second-order Hamiltonian systems by using the variant fountain theorem.

1 Introduction

Consider the second-order Hamiltonian systems

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M2">View MathML</a> is also T-periodic and satisfies the following assumption (A):

(A) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M2">View MathML</a> is measurable in t for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M4">View MathML</a>, continuously differentiable in u for a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M5">View MathML</a> and there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M6">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M7">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M8">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M4">View MathML</a> and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M5">View MathML</a>.

Here and in the sequel, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M11">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M12">View MathML</a> always denote the standard inner product and the norm in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M13">View MathML</a> respectively.

There have been many investigations on the existence and multiplicity of periodic solutions for Hamiltonian systems via the variational methods (see [1-7] and the references therein). In [6], Zhang and Liu studied the asymptotically quadratic case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M14">View MathML</a> under the following assumptions:

(AQ1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M15">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M16">View MathML</a>, and there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M18">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M19">View MathML</a>

(AQ2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M20">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M21">View MathML</a>, and there exist constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M22">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M23">View MathML</a>

(AQ3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M24">View MathML</a> uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M5">View MathML</a>.

They obtained the existence of infinitely many periodic solutions of (1.1) provided <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M26">View MathML</a>is even inu (see Theorem 1.1 of [6]).

The subquadratic condition (AQ1) is widely used in the investigation of nonlinear differential equations. This condition was weakened by some researchers; see, for example, [4] of Jiang and Tang. This paper considers the case of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M27">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M28">View MathML</a>. Motivated by [4] and [6], we replace (AQ1) with the following condition:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M29">View MathML</a>) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M30">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M16">View MathML</a>, and

The condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M33">View MathML</a>) implies that for some constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M34">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M35">View MathML</a>

(1.2)

By the assumption (A) and the condition (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M33">View MathML</a>), for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M37">View MathML</a>, there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M38">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M39">View MathML</a>

(1.3)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M40">View MathML</a> and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M5">View MathML</a>.

Meanwhile, we weaken the condition (AQ3) to (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M42">View MathML</a>) as follows:

(<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M42">View MathML</a>) There exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M44">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M45">View MathML</a>

Then our main result is the following theorem.

Theorem 1.1Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M29">View MathML</a>), (AQ2), (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M42">View MathML</a>) hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M2">View MathML</a>is even inu. Then (1.1) possesses infinitely many solutions.

Remark The conditions (AQ1) and (AQ3) are stronger than (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M29">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M42">View MathML</a>). Then Theorem 1.1 above is different from Theorem 1.1 of [6].

2 Preliminaries

In this section, we establish the variational setting for our problem and give the variant fountain theorem. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M51">View MathML</a> be the usual Sobolev space with the inner product

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M52">View MathML</a>

We define the functional on E by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M53">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M54">View MathML</a>. Then Φ and Ψ are continuously differentiable and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M55">View MathML</a>

Define a self-adjoint linear operator <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M56">View MathML</a> by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M57">View MathML</a>

with the domain <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M58">View MathML</a>. Then ℬ has a sequence of eigenvalues <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M59">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M60">View MathML</a>). Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M61">View MathML</a> be the system of eigenfunctions corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M62">View MathML</a>, it forms an orthogonal basis in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M63">View MathML</a>. Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M65">View MathML</a>, it is well known that

and E possesses orthogonal decomposition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M67">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M68">View MathML</a>, we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M69">View MathML</a>

We can define on E a new inner product and the associated norm by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M70">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M71">View MathML</a>

Therefore, Φ can be written as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M72">View MathML</a>

(2.1)

Direct computation shows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M73">View MathML</a>

(2.2)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M74">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M75">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M76">View MathML</a> respectively. It is known that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M77">View MathML</a> is compact.

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M78">View MathML</a> the usual norm of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M79">View MathML</a>, then there exists a <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M80">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M81">View MathML</a>

(2.3)

We state an abstract critical point theorem founded in [8]. Let E be a Banach space with the norm <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M82">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M83">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M84">View MathML</a> for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M85">View MathML</a>. Set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M86">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M87">View MathML</a> . Consider the following <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M88">View MathML</a>-functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M89">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M90">View MathML</a>

Theorem 2.1 [[8], Theorem 2.2]

Assume that the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M91">View MathML</a>defined above satisfies the following:

(T1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M91">View MathML</a>maps bounded sets to bounded sets uniformly for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M93">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M94">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M95">View MathML</a>;

(T2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M96">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M68">View MathML</a>, and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M98">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M99">View MathML</a>on any finite-dimensional subspace ofE;

(T3) There exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M100">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M101">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M102">View MathML</a>

Then there exist<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M103">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M104">View MathML</a>such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M105">View MathML</a>

Particularly, if<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M106">View MathML</a>has a convergent subsequence for everyk, then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M107">View MathML</a>has infinitely many nontrivial critical points<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M108">View MathML</a>satisfying<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M109">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M110">View MathML</a>.

In order to apply this theorem to prove our main result, we define the functionals A, B and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M91">View MathML</a> on our working space E by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M112">View MathML</a>

(2.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M113">View MathML</a>

(2.5)

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M114">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M93">View MathML</a>. Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M116">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M93">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M118">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M119">View MathML</a> . Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M120">View MathML</a>, where Φ is the functional defined in (2.1).

3 Proof of Theorem 1.1

We firstly establish the following lemmas.

Lemma 3.1Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M33">View MathML</a>) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M122">View MathML</a>) hold. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M96">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M68">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M125">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M99">View MathML</a>on any finite-dimensional subspace of E.

Proof Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M30">View MathML</a>, by (2.4), it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M96">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M68">View MathML</a>.

By the proof of Lemma 2.6 of [6], for any finite-dimensional subspace <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M130">View MathML</a>, there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M37">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M132">View MathML</a>

(3.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M133">View MathML</a> is the Lebesgue measure.

For the ϵ given in (3.1), let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M134">View MathML</a>

Then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M135">View MathML</a>. By (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M122">View MathML</a>), there exists a constant <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M137">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M138">View MathML</a>

(3.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M139">View MathML</a> is the constant given in (1.2). Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M140">View MathML</a>

(3.3)

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M141">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M142">View MathML</a>. Thus,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M143">View MathML</a>

for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M141">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M142">View MathML</a>. This implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M125">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M99">View MathML</a> on Y. □

Lemma 3.2Assume that (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M33">View MathML</a>), (AQ2) and (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M122">View MathML</a>) hold. Then there exist a positive integer<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M150">View MathML</a>and two sequences<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M151">View MathML</a>as<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M110">View MathML</a>such that

(3.4)

(3.5)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M155">View MathML</a>

(3.6)

where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M156">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M157">View MathML</a>for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M158">View MathML</a>.

Proof Comparing this lemma with Lemma 2.7 of [6], we find that these two lemmas have the same condition (AQ2) which is the key in the proof of Lemma 2.7 of [6]. We can prove our lemma by using the same method of [6], so the details are omitted. □

Now it is the time to prove our main result Theorem 1.1.

Proof of Theorem 1.1 By virtue of (1.3), (2.3) and (2.5), <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M91">View MathML</a> maps bounded sets to bounded sets uniformly for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M93">View MathML</a>. Obviously, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M94">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M162">View MathML</a> since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M2">View MathML</a> is even in u. Consequently, the condition (T1) of Theorem 2.1 holds. Lemma 3.1 shows that the condition (T2) holds, whereas Lemma 3.2 implies that the condition (T3) holds for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M164">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M150">View MathML</a> is given there. Therefore, by Theorem 2.1, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M164">View MathML</a>, there exist <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M103">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M104">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M169">View MathML</a>

(3.7)

For the sake of notational simplicity, in the following we always set <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M170">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M171">View MathML</a>.

Step 1. We firstly prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M172">View MathML</a> is bounded in E.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M172">View MathML</a> satisfies (3.7), one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M174">View MathML</a>

More precisely,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M175">View MathML</a>

(3.8)

Now, we prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M172">View MathML</a> is bounded. Otherwise, without loss of generality, we may assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M177">View MathML</a>

Put <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M178">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M179">View MathML</a>. Going to a subsequence if necessary, we may assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M180">View MathML</a>

By (1.3), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M181">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M182">View MathML</a>. Therefore, one obtains

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M183">View MathML</a>

Passing to the limit in the inequality, by using <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M184">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M103">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M186">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M187">View MathML</a>

Thus, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M188">View MathML</a> on a subset Ω of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M189">View MathML</a> with positive measure.

By (1.2), we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M190">View MathML</a>

and by the assumption (A), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M191">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M192">View MathML</a>. So, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M193">View MathML</a>

for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M4">View MathML</a> and a.e. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M5">View MathML</a>. Hence,

An application of Fatou’s lemma yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M197">View MathML</a>

which is a contradiction to (3.8).

Step 2. We prove that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M172">View MathML</a> has a convergent subsequence in E.

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M172">View MathML</a> is bounded in E, E is reflexible and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M200">View MathML</a>, without loss of generality, we assume

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M201">View MathML</a>

(3.9)

for some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M202">View MathML</a>.

Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M203">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M204">View MathML</a> is the orthogonal projection for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M205">View MathML</a>, that is,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M206">View MathML</a>

(3.10)

In view of the compactness of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M207">View MathML</a> and (3.9), the right-hand side of (3.10) converges strongly in E and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M208">View MathML</a> in E. Together with (3.9), we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M209">View MathML</a> in E.

Now, from the last assertion of Theorem 2.1, we know that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M210','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/16/mathml/M210">View MathML</a> has infinitely many nontrivial critical points. The proof is completed. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

HG wrote the first draft and TA corrected and improved the final version. All authors read and approved the final draft.

Acknowledgements

The authors thank the referee for his/her careful reading of the manuscript. The work is supported by the Fundamental Research Funds for the Central Universities and the National Natural Science Foundation of China (No. 61001139).

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