Open Access Research

Solutions to a boundary value problem of a fourth-order impulsive differential equation

Jingli Xie1 and Zhiguo Luo2*

Author Affiliations

1 College of Mathematics and Statistics, Jishou University, Jishou, Hunan, 416000, P.R. China

2 Department of Mathematics, Hunan Normal University, Changsha, Hunan, 410081, P.R. China

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Boundary Value Problems 2013, 2013:154  doi:10.1186/1687-2770-2013-154


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/154


Received:26 March 2013
Accepted:15 June 2013
Published:1 July 2013

© 2013 Xie and Luo; 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

This paper is concerned with the existence of solutions to a boundary value problem of a fourth-order impulsive differential equation with a control parameter λ. By employing some existing critical point theorems, we find the range of the control parameter in which the boundary value problem admits at least one solution. It is also shown that under certain conditions there exists an interval of the control parameter in which the boundary value problem possesses infinitely many solutions. The main results are also demonstrated with examples.

MSC: 34B15, 34B18, 34B37, 58E30.

Keywords:
critical point theorem; impulsive differential equations; boundary value problem

1 Introduction

Fourth-order two-point boundary value problems of ordinary differential equations are widely employed by engineers to describe the beam deflection with two simply supported ends [1-3]. One example is the following fourth-order two-point boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M1">View MathML</a>

(1.1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M2">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M4">View MathML</a> are the fourth, third, and second derivatives of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M5">View MathML</a> with respect to t, respectively, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M6">View MathML</a>, A and B are two real constants. System (1.1) has been studied in [4-7] and the references therein. For a beam, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M8">View MathML</a> in (1.1) refer to the two ends of the beam. At other locations of the beam, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M9">View MathML</a>, there may be some sudden changes in loads placed on the beam, or some unexpected forces working on the beam. These sudden changes may result in impulsive effects for the governing differential equation. This motivates us to consider the following boundary value problem for a fourth-order impulsive differential equation:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M10">View MathML</a>

(1.2)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M12">View MathML</a>, and the operator Δ is defined as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M13">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M14">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M15">View MathML</a>) denotes the right-hand (left-hand) limit of U at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M16">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M17">View MathML</a> is referred to as a control parameter.

We are mainly concerned with the existence of solutions of system (1.2). A function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M18">View MathML</a> is said to be a (classical) solution of (1.2) if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M5">View MathML</a> satisfies (1.2). In literature, tools employed to establish the existence of solutions of impulsive differential equations include fixed point theorems, the upper and lower solutions method, the degree theory, critical point theory and variational methods. See, for example, [8-20]. In this paper, we establish the existence of solutions of (1.2) by converting the problem to the existence of critical points of some variational structure. In this paper we regard λ as a parameter and find the ranges in which (1.2) admits at least one and infinitely many solutions, respectively. Note that when <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M20">View MathML</a> system (1.2) reduces to the one studied in [21]. Our results extend those ones in [21].

The rest of this paper is organized as follows. In Section 2 we present some preliminary results. Our main results and their proofs are given in Section 3.

2 Preliminaries

Throughout we assume that A and B satisfy

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M21">View MathML</a>

(2.1)

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M22">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M23">View MathML</a>

Take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M24">View MathML</a> and define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M25">View MathML</a>

(2.2)

Since A and B satisfy (2.1), it is straightforward to verify that (2.2) defines a norm for the Sobolev space X and this norm is equivalent to the usual norm defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M26">View MathML</a>

It follows from (2.1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M27">View MathML</a>

For the norm in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M28">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M29">View MathML</a>

we have the following relation.

Lemma 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M30">View MathML</a>. Then<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M32">View MathML</a>.

Proof The proof follows easily from Wirtinger’s inequality [22], Lemma 2.3 of [23] and Hölder’s inequality. The detailed argument is similar to the proof of Lemma 2.2 in [21], and we thus omit it here. □

Define a functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M33">View MathML</a> as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M34">View MathML</a>

(2.3)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M35">View MathML</a>

(2.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M36">View MathML</a>

(2.5)

with

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M37">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M33">View MathML</a> is Fréchet differentiable at any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M39">View MathML</a>, and for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M40">View MathML</a> we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M41">View MathML</a>

(2.6)

Next we show that a critical point of the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M33">View MathML</a> is a solution of system (1.2).

Lemma 2.2If<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M39">View MathML</a>is a critical point of<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M33">View MathML</a>, thenuis a solution of system (1.2).

Proof Suppose that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M39">View MathML</a> is a critical point of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M46">View MathML</a>. Then for any <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M40">View MathML</a> one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M48">View MathML</a>

(2.7)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M49">View MathML</a>, choose <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M40">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M51">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M52">View MathML</a>, then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M53">View MathML</a>

Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M54">View MathML</a>

Therefore, by (2.7) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M55">View MathML</a>

Next we show that u satisfies

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M56">View MathML</a>

Suppose on the contrary that there exists some <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M57">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M58">View MathML</a>

Pick

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M59">View MathML</a>

then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M60">View MathML</a>

Clearly, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M40">View MathML</a>. Simple calculations show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M62">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M64">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M66">View MathML</a>. Thus

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M67">View MathML</a>

which is a contradiction. Similarly, one can show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M69">View MathML</a>. Therefore, u is a solution of (1.2). □

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M70">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M71">View MathML</a>, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M72">View MathML</a>

(2.8)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M73">View MathML</a>

(2.9)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M74">View MathML</a>, we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M75">View MathML</a>

(2.10)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M76">View MathML</a>

(2.11)

3 Main results

3.1 Existence of at least one solution

In this section we derive conditions under which system (1.2) admits at least one solution. For this purpose, we introduce the following assumption.

(H1) Assume that there exist two positive constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M77">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M78">View MathML</a> such that for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M39">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M80">View MathML</a>

(3.1)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M81">View MathML</a>

(3.2)

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M82">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M83">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M77">View MathML</a> given in (3.1) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M78">View MathML</a> given in (3.2). For constants <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M87">View MathML</a>, and c satisfying

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M88">View MathML</a>

(3.3)

we define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M89">View MathML</a>

(3.4)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M90">View MathML</a>

(3.5)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M91">View MathML</a>

(3.6)

Note that for every <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M92">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M93">View MathML</a> we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M94">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M95">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M96">View MathML</a>. Thus, if c and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M87">View MathML</a> satisfy (3.3), then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M98">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M99">View MathML</a> and hence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M100">View MathML</a>.

Theorem 3.1Assume that (H1) is satisfied. If there exist constants<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M86">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M87">View MathML</a>, andcsatisfying (3.3) and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M103">View MathML</a>

(3.7)

then, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M104">View MathML</a>, system (1.2) admits at least one solutionuand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M105">View MathML</a>, where<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M106">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M107">View MathML</a>.

Proof By Lemma 2.2, it suffices to show the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M33">View MathML</a> defined in (2.3) has at least one critical point. We prove this by verifying the conditions given in [[10], Theorem 5.1]. Note that Φ defined in (2.4) is a nonnegative Gâteaux differentiable, coercive, and sequentially weakly lower semicontinuous functional, and its Gâteaux derivative admits a continuous inverse on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M109">View MathML</a>. Moreover, Ψ defined in (2.5) is a continuously Gâteaux differentiable functional whose Gâteaux derivative is compact. Set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M110">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M111">View MathML</a>. It then follows from (H1) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M112">View MathML</a>

(3.8)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M113">View MathML</a>

(3.9)

By (3.3) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M114">View MathML</a>

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M39">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M116">View MathML</a>, by Lemma 2.1, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M117">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M118">View MathML</a>

Hence

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M119">View MathML</a>

(3.10)

For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M39">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M121">View MathML</a>, one can similarly obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M122">View MathML</a>

(3.11)

It follows from the definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M123">View MathML</a> that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M124">View MathML</a>

(3.12)

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M125">View MathML</a>. By (3.10) one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M126">View MathML</a>

Making use of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M125">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M128">View MathML</a>, and (3.8), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M129">View MathML</a>

By (2.8), and note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M130">View MathML</a>, one has

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M131">View MathML</a>

By (3.11) we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M132">View MathML</a>

Note that (3.7) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M133">View MathML</a>

which, together with (3.8), gives

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M134">View MathML</a>

Therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M135">View MathML</a>. Thus all the conditions in [[10], Theorem 5.1] are verified, and hence for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M136">View MathML</a> the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M137">View MathML</a> admits at least one critical point u such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M138">View MathML</a>. Consequently, system (1.2) admits at least one solution u and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M105">View MathML</a>. □

In particular, if we take <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M140">View MathML</a>, then (3.4) and (3.5) become

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M141">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M142">View MathML</a>

Correspondingly, conditions (3.3) and (3.7) reduce to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M143">View MathML</a>

(3.13)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M144">View MathML</a>

(3.14)

If (3.13) and (3.14) hold, then

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M145">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M146">View MathML</a>

As a consequence, we have the following result.

Corollary 3.2Assume that (H1) is satisfied. If there exist two constantscand<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M87">View MathML</a>satisfying (3.13) and (3.14), then for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M148">View MathML</a>system (1.2) admits at least one nontrivial solutionu.

Example 3.1 Consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M149">View MathML</a>

(3.15)

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M151">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M152">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M153">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M154">View MathML</a>. It is easy to verify that (H1) is satisfied with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M155">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M156">View MathML</a>. Direct calculations give <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M158">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M159">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M160">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M161">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M162">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M163">View MathML</a>, then <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M86">View MathML</a>, c, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M87">View MathML</a> satisfy (3.3) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M166">View MathML</a>. Thus, it follows from Theorem 3.1 that system (3.15) has at least one solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M167">View MathML</a>.

3.2 Existence of infinitely many solutions

In this section, we derive some conditions under which system (1.2) admits infinitely many distinct solutions. To this end, we need the following assumptions.

(H2) Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M168">View MathML</a>

(H3) Assume that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M169">View MathML</a>

Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M170">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M171">View MathML</a> with <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M78">View MathML</a> given in (3.2). We define

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M173">View MathML</a>

(3.16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M174">View MathML</a> is given (2.10). Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M175">View MathML</a>

Theorem 3.3Assume that (H1), (H2), and (H3) are satisfied. If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M176">View MathML</a>

(3.17)

holds, then for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M177">View MathML</a>system (1.2) has an unbounded sequence of solutions inX.

Proof We apply [[5], Theorem 2.1] to show that the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M33">View MathML</a> defined in (2.3) has an unbounded sequence of critical points.

We first show that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M179">View MathML</a>. Let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M180','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M180">View MathML</a> be a sequence of positive numbers such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M181','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M181">View MathML</a> as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M182','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M182">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M183">View MathML</a>

For any positive integer n, we let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M184">View MathML</a>. For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M39">View MathML</a> satisfying <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M186','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M186">View MathML</a>, similar to the proof of Theorem 3.1, one can show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M187','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M187">View MathML</a>

which implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M188">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M189','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M189">View MathML</a>, thus we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M190">View MathML</a>

which, together with (3.16), gives us

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M191','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M191">View MathML</a>

This shows that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M192">View MathML</a>. For any fixed <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M193">View MathML</a>, it follows from [[5], Theorem 2.1] that either <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M137">View MathML</a> has a global minimum or there is a sequence <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M195">View MathML</a> of critical points (local minima) of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M46">View MathML</a> such that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M197">View MathML</a>.

Next we show that the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M33">View MathML</a> has no global minimum for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M193">View MathML</a>. Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M200">View MathML</a>, we can choose a constant M such that, for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M201">View MathML</a>,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M202">View MathML</a>

Thus, there exists <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M203','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M203">View MathML</a> such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M204">View MathML</a>

Define <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M205','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M205">View MathML</a> as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M206">View MathML</a>

This, together with (H2), yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M207','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M207">View MathML</a>

It then follows from (H3) that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M208">View MathML</a>

Note that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M209','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M209">View MathML</a>. Thus the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M33">View MathML</a> is unbounded from below and hence it has no global minimum and the proof is complete. □

Corollary 3.4Assume that (H1), (H2), and (H3) are satisfied. If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M211">View MathML</a>

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M212','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M212">View MathML</a>

hold, then (1.2) has an unbounded sequence of solutions inX.

Let

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M213">View MathML</a>

Theorem 3.5Assume that (H1), (H2), and (H3) are satisfied. If

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M214','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M214">View MathML</a>

(3.18)

holds, then for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M215','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M215">View MathML</a>system (1.2) has a sequence of non-zero solutions inX, which weakly converges to 0.

Proof The proof is similar to that of Theorem 3.3 by showing that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M216','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M216">View MathML</a> and 0 is not a local minimum of the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M217">View MathML</a>. □

Example 3.2 Consider

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M218','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M218">View MathML</a>

(3.19)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M219">View MathML</a>.

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M220">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M221">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M222">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M223">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M154">View MathML</a>. Note that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M225">View MathML</a>

so (H1), (H2), and (H3) are satisfied. Moreover, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M226">View MathML</a>, and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M227','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M227">View MathML</a>

Therefore, condition (3.17) holds and Theorem 3.3 applies: For <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M228">View MathML</a>, (3.19) admits an unbounded sequence of solutions in X.

Example 3.3 Consider the boundary value problem

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M229">View MathML</a>

(3.20)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M230','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M230">View MathML</a>

In this example, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M231">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M232">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M222','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M222">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M223','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M223">View MathML</a>. The assumptions (H1), (H2), and (H3) clearly hold.

Direct calculations give

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M235">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M236">View MathML</a> and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M237">View MathML</a>

Hence (3.18) holds. Therefore it follows from Theorem 3.5 that (3.20) admits a sequence of distinct solutions in X provided that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M238','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/154/mathml/M238">View MathML</a>.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors made equal contribution. Both authors read and approved the final manuscript.

Authors’ information

JX is with the College of Mathematics and Statistics, Jishou University, China and is a PhD candidate at the Department of Mathematics, Hunan Normal University, China. ZL is a professor at the Department of Mathematics, Hunan Normal University, China.

Acknowledgements

The authors are very grateful to the referees for their valuable comments and suggestions, which greatly improved the presentation of this paper. The work is partially supported by Hunan Provincial Natural Science Foundation of China (No: 11JJ3012).

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