Open Access

Variational Approach to Impulsive Differential Equations with Dirichlet Boundary Conditions

Boundary Value Problems20102010:325415

DOI: 10.1155/2010/325415

Received: 18 September 2010

Accepted: 9 November 2010

Published: 24 November 2010

Abstract

We study the existence of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq1_HTML.gif distinct pairs of nontrivial solutions for impulsive differential equations with Dirichlet boundary conditions by using variational methods and critical point theory.

1. Introduction

Impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time. Such processes are naturally seen in control theory [1, 2], population dynamics [3], and medicine [4, 5]. Due to its significance, a great deal of work has been done in the theory of impulsive differential equations. In recent years, many researchers have used some fixed point theorems [6, 7], topological degree theory [8], and the method of lower and upper solutions with monotone iterative technique [9] to study the existence of solutions for impulsive differential equations.

On the other hand, in the last few years, some researchers have used variational methods to study the existence of solutions for boundary value problems [1016], especially, in [1416], the authors have studied the existence of infinitely many solutions by using variational methods.

However, as far as we know, few researchers have studied the existence of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq2_HTML.gif distinct pairs of nontrivial solutions for impulsive boundary value problems by using variational methods.

Motivated by the above facts, in this paper, our aim is to study the existence of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq3_HTML.gif distinct pairs of nontrivial solutions to the Dirichlet boundary problem for the second-order impulsive differential equations
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq4_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq6_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq7_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq9_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq10_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq11_HTML.gif denote the right and the left limits, respectively, of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq12_HTML.gif at https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq13_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq14_HTML.gif .

2. Preliminaries

Definition 2.1.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq15_HTML.gif is a Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq16_HTML.gif . If any sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq17_HTML.gif for which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq18_HTML.gif is bounded and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq19_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq20_HTML.gif possesses a convergent subsequence in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq21_HTML.gif , we say that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq22_HTML.gif satisfies the Palais-Smale condition.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq23_HTML.gif be a real Banach space. Define the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq24_HTML.gif as symmetric closed set}.

Theorem 2.2 (see [17, Theorem 3.5.3]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq25_HTML.gif be a real Banach space, and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq26_HTML.gif be an even functional which satisfies the Palais-Smale condition, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq27_HTML.gif is bounded from below and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq28_HTML.gif ; suppose that there exists a set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq29_HTML.gif and an odd homeomorphism https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq30_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq31_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq32_HTML.gif has at least n distinct pairs of nontrivial critical points.

To begin with, we introduce some notation. Denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq33_HTML.gif the Sobolev space https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq34_HTML.gif , and consider the inner product
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ2_HTML.gif
(2.1)
and the norm
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ3_HTML.gif
(2.2)

Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq35_HTML.gif is reflexive. We define the norm in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq36_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq37_HTML.gif .

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq38_HTML.gif , we have that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq39_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq40_HTML.gif are absolutely continuous and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq41_HTML.gif . Hence, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq42_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq43_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq44_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq45_HTML.gif is absolutely continuous and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq46_HTML.gif . In this case, the one-sided derivatives https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq47_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq48_HTML.gif may not exist. As a consequence, we need to introduce a different concept of solution. Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq49_HTML.gif such that for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq50_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq51_HTML.gif satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq52_HTML.gif , and it satisfies the equation in problem (1.1) for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq53_HTML.gif , a.e. https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq54_HTML.gif , the limits https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq55_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq56_HTML.gif exist, and impulsive conditions and boundary conditions in problem (1.1) hold, we say it is a classical solution of problem (1.1).

Consider the functional
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ4_HTML.gif
(2.3)
defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ5_HTML.gif
(2.4)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq57_HTML.gif . Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq58_HTML.gif is a Fréchet differentiable functional, whose Fréchet derivative at the point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq59_HTML.gif is the functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq60_HTML.gif given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ6_HTML.gif
(2.5)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq61_HTML.gif . Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq62_HTML.gif is continuous.

Lemma 2.3.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq63_HTML.gif is a critical point of the functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq64_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq65_HTML.gif is a classical solution of problem (1.1).

Proof.

The proof is similar to the proof of [16, Lemma 2.4], and we omit it here.

Lemma 2.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq66_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq67_HTML.gif .

Proof.

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq68_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq69_HTML.gif . Hence, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq70_HTML.gif , by Hölder's inequality, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ7_HTML.gif
(2.6)

which completes the proof.

3. Main Results

Theorem 3.1.

Suppose that the following conditions hold.

(i) There exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq71_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq72_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ8_HTML.gif
(3.1)

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq73_HTML.gif is odd about u and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq74_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq75_HTML.gif .

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq76_HTML.gif are odd and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq77_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq78_HTML.gif .

Then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq79_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq80_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq81_HTML.gif , and problem (1.1) has at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq82_HTML.gif distinct pairs of nontrivial classical solutions.

Proof.

By (2.4), (ii), and (iii), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq83_HTML.gif is an even functional and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq84_HTML.gif .

Next, we will verify that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq85_HTML.gif is bounded from below. In view of (i), (iii), and Lemma 2.4, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ9_HTML.gif
(3.2)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq86_HTML.gif . That is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq87_HTML.gif is bounded from below.

In the following we will show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq88_HTML.gif satisfies the Palais-Smale condition. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq89_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq90_HTML.gif is a bounded sequence and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq91_HTML.gif . Then, there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq92_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ10_HTML.gif
(3.3)
In view of (3.2), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ11_HTML.gif
(3.4)
So https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq93_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq94_HTML.gif . From the reflexivity of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq95_HTML.gif , we may extract a weakly convergent subsequence that, for simplicity, we call https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq96_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq97_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq98_HTML.gif . Next, we will verify that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq99_HTML.gif strongly converges to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq100_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq101_HTML.gif . By (2.5), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ12_HTML.gif
(3.5)
By https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq102_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq103_HTML.gif , we see that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq104_HTML.gif uniformly converges to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq105_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq106_HTML.gif . So,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ13_HTML.gif
(3.6)
By https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq107_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq108_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ14_HTML.gif
(3.7)

In view of (3.5), (3.6), and (3.7), we obtain https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq109_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq110_HTML.gif satisfies the Palais-Smale condition.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq111_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq112_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ15_HTML.gif
(3.8)
Define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ16_HTML.gif
(3.9)
Then, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq113_HTML.gif , there exists an odd homeomorphism https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq114_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq115_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq116_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq117_HTML.gif . By (ii), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ17_HTML.gif
(3.10)

then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq118_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq119_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq120_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq121_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq122_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq123_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq124_HTML.gif , then when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq125_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq126_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ18_HTML.gif
(3.11)

By Theorem 2.2, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq127_HTML.gif possesses at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq128_HTML.gif distinct pairs of nontrivial critical points. That is, problem (1.1) has at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq129_HTML.gif distinct pairs of nontrivial classical solutions.

Corollary 3.2.

Let the following conditions hold:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq130_HTML.gif is bounded,

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq131_HTML.gif is odd about u and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq132_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq133_HTML.gif ,

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq134_HTML.gif are odd and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq135_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq136_HTML.gif .

Then, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq137_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq138_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq139_HTML.gif , and problem (1.1) has at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq140_HTML.gif distinct pairs of nontrivial classical solutions.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq141_HTML.gif in Theorem 3.1, then Corollary 3.2 holds.

Theorem 3.3.

Suppose that the following conditions hold.

(i) There exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq142_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq143_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ19_HTML.gif
(3.12)
(ii) There exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq144_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq145_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ20_HTML.gif
(3.13)

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq146_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq147_HTML.gif are odd about u and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq148_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq149_HTML.gif .

Then, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq150_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq151_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq152_HTML.gif , and problem (1.1) has at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq153_HTML.gif distinct pairs of nontrivial classical solutions.

Proof.

By (2.4) and (iii), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq154_HTML.gif is an even functional and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq155_HTML.gif .

Next, we will verify that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq156_HTML.gif is bounded from below. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq157_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq158_HTML.gif . In view of (i), (ii), and Lemma 2.4, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ21_HTML.gif
(3.14)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq159_HTML.gif . That is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq160_HTML.gif is bounded from below.

In the following, we will show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq161_HTML.gif satisfies the Palais-Smale condition. As in the proof of Theorem 3.1, by (3.3) and (3.14), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ22_HTML.gif
(3.15)

It follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq162_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq163_HTML.gif . In the following, the proof of the Palais-Smale condition is the same as that in Theorem 3.1, and we omit it here.

Take the same https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq164_HTML.gif as in Theorem 3.1, then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq165_HTML.gif , there exists an odd homeomorphism https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq166_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq167_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq168_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq169_HTML.gif . By (iii), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ23_HTML.gif
(3.16)

Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq170_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq171_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq172_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq173_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq174_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq175_HTML.gif , then when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq176_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq177_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ24_HTML.gif
(3.17)

By Theorem 2.2, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq178_HTML.gif possesses at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq179_HTML.gif distinct pairs of nontrivial critical points. That is, problem (1.1) has at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq180_HTML.gif distinct pairs of nontrivial classical solutions.

Corollary 3.4.

Let the following conditions hold:

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq181_HTML.gif is bounded,

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq182_HTML.gif are bounded,

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq183_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq184_HTML.gif are odd about u and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq185_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq186_HTML.gif .

Then, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq187_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq188_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq189_HTML.gif , and problem (1.1) has at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq190_HTML.gif distinct pairs of nontrivial classical solutions.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq191_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq192_HTML.gif in Theorem 3.3, then Corollary 3.4 holds.

Theorem 3.5.

Suppose that the following conditions hold.

(i) There exist constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq193_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq194_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq195_HTML.gif .

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq196_HTML.gif is odd about https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq197_HTML.gif .

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq198_HTML.gif are odd and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq199_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq200_HTML.gif .

Then, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq201_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq202_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq203_HTML.gif , and problem (1.1) has at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq204_HTML.gif distinct pairs of nontrivial classical solutions.

Proof.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ25_HTML.gif
(3.18)
then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq205_HTML.gif is continuous, bounded, and odd. Consider boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ26_HTML.gif
(3.19)
Next, we will verify that the solutions of problem (3.19) are solutions of problem (1.1). In fact, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq206_HTML.gif be the solution of problem (3.19). If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq207_HTML.gif , then there exists an interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq208_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ27_HTML.gif
(3.20)
When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq209_HTML.gif , by (i), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ28_HTML.gif
(3.21)

Thus, there exist constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq210_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq211_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq212_HTML.gif . We consider the following two possible cases.

Case 1.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq213_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq214_HTML.gif is nondecreasing in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq215_HTML.gif . By https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq216_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq217_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ29_HTML.gif
(3.22)

That is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq218_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq219_HTML.gif . So, there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq220_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq221_HTML.gif , which contradicts (3.20). Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq222_HTML.gif . Similarly, we can prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq223_HTML.gif .

Case 2.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq224_HTML.gif , the arguments are analogous, then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq225_HTML.gif is solution of problem (1.1).

For every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq226_HTML.gif , we consider the functional
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ30_HTML.gif
(3.23)
defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ31_HTML.gif
(3.24)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq227_HTML.gif .

It is clear that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq228_HTML.gif is Fréchet differentiable at any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq229_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ32_HTML.gif
(3.25)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq230_HTML.gif . Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq231_HTML.gif is continuous. By Lemma  2.3, we have the critical points of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq232_HTML.gif as solutions of problem (3.19). By (3.24), (ii), and (iii), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq233_HTML.gif is an even functional and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq234_HTML.gif .

In the following, we will show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq235_HTML.gif is bounded from below. since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq236_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq237_HTML.gif , thus
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ33_HTML.gif
(3.26)
By (iii), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ34_HTML.gif
(3.27)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq238_HTML.gif . That is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq239_HTML.gif is bounded from below.

In the following we will show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq240_HTML.gif satisfies the Palais-Smale condition. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq241_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq242_HTML.gif is a bounded sequence and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq243_HTML.gif . Then, there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq244_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ35_HTML.gif
(3.28)
By (3.27), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ36_HTML.gif
(3.29)

It follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq245_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq246_HTML.gif . In the following, the proof of the Palais-Smale condition is the same as that in Theorem 3.1, and we omit it here.

Take the same https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq247_HTML.gif as in Theorem 3.1, then, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq248_HTML.gif , there exists an odd homeomorphism https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq249_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq250_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq251_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq252_HTML.gif . By (i) and (ii), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ37_HTML.gif
(3.30)

Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq253_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq254_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq255_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq256_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq257_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq258_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq259_HTML.gif , then when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq260_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq261_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ38_HTML.gif
(3.31)

By Theorem 2.2, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq262_HTML.gif possesses at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq263_HTML.gif distinct pairs of nontrivial critical points. Then, problem (3.19) has at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq264_HTML.gif distinct pairs of nontrivial classical solutions, that is, problem (1.1) has at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq265_HTML.gif distinct pairs of nontrivial classical solutions

Theorem 3.6.

Let the following conditions hold.

(i) There exist constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq266_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq267_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq268_HTML.gif .

(ii) There exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq269_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq270_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ39_HTML.gif
(3.32)

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq271_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq272_HTML.gif are odd about https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq273_HTML.gif .

Then, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq274_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq275_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq276_HTML.gif , and problem (1.1) has at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq277_HTML.gif distinct pairs of nontrivial classical solutions.

Proof.

The proof is similar to the proof of Theorem 3.5, and we omit it here.

Theorem 3.7.

Let the following conditions hold.

(i) There exist constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq278_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq279_HTML.gif .

(ii) There exist https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq280_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq281_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ40_HTML.gif
(3.33)

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq282_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq283_HTML.gif are odd about https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq284_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq285_HTML.gif uniformly for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq286_HTML.gif .

Then, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq287_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq288_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq289_HTML.gif , and problem (1.1) has at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq290_HTML.gif distinct pairs of nontrivial classical solutions.

Proof.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ41_HTML.gif
(3.34)
then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq291_HTML.gif is continuous, bounded, and odd. Consider boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ42_HTML.gif
(3.35)
Next, we will verify that the solutions of problem (3.35) are solutions of problem (1.1). In fact, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq292_HTML.gif be the solution of problem (3.35). If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq293_HTML.gif , then there exists an interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq294_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ43_HTML.gif
(3.36)
When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq295_HTML.gif , by (i), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ44_HTML.gif
(3.37)
Thus, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq296_HTML.gif is nondecreasing in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq297_HTML.gif . By https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq298_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq299_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ45_HTML.gif
(3.38)

That is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq300_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq301_HTML.gif . So, there exists a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq302_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq303_HTML.gif , which contradicts (3.36). Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq304_HTML.gif . Similarly, we can prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq305_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq306_HTML.gif is solution of problem (1.1).

For every https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq307_HTML.gif , we consider the functional
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ46_HTML.gif
(3.39)
defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ47_HTML.gif
(3.40)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq308_HTML.gif .

It is clear that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq309_HTML.gif is Fréchet differentiable at any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq310_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ48_HTML.gif
(3.41)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq311_HTML.gif . Obviously, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq312_HTML.gif is continuous. By Lemma 2.3, we have the critical points of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq313_HTML.gif as solutions of problem (3.35). By (3.40) and (iii), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq314_HTML.gif is an even functional and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq315_HTML.gif .

Next, we will show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq316_HTML.gif is bounded from below. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq317_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq318_HTML.gif . since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq319_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq320_HTML.gif , thus
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ49_HTML.gif
(3.42)
By (ii) and Lemma 2.4, we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ50_HTML.gif
(3.43)

for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq321_HTML.gif . That is, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq322_HTML.gif is bounded from below.

In the following we will show that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq323_HTML.gif satisfies the Palais-Smale condition. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq324_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq325_HTML.gif is a bounded sequence and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq326_HTML.gif . Then, there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq327_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ51_HTML.gif
(3.44)
By (3.43), we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ52_HTML.gif
(3.45)

It follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq328_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq329_HTML.gif . In the following, the proof of the Palais-Smale condition is the same as that in Theorem 3.1, and we omit it here.

Take the same https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq330_HTML.gif as in Theorem 3.1, then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq331_HTML.gif , there exists an odd homeomorphism https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq332_HTML.gif . By (iii), for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq333_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq334_HTML.gif , when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq335_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ53_HTML.gif
(3.46)

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq336_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq337_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq338_HTML.gif . Then, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq339_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq340_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq341_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq342_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq343_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq344_HTML.gif , then when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq345_HTML.gif , for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq346_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ54_HTML.gif
(3.47)

By Theorem 2.2, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq347_HTML.gif possesses at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq348_HTML.gif distinct pairs of nontrivial critical points. Then, problem (3.35) has at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq349_HTML.gif distinct pairs of nontrivial classical solutions, that is, problem (1.1) has at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq350_HTML.gif distinct pairs of nontrivial classical solutions.

Theorem 3.8.

Let the following conditions hold.

(i) There exist constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq351_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq352_HTML.gif .

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq353_HTML.gif uniformly for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq354_HTML.gif .

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq355_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq356_HTML.gif are odd about https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq357_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq358_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq359_HTML.gif .

Then, for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq360_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq361_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq362_HTML.gif , and problem (1.1) has at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq363_HTML.gif distinct pairs of nontrivial classical solutions.

Proof.

The proof is similar to the proof of Theorem 3.7, and we omit it here.

4. Some Examples

Example 4.1.

Consider boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ55_HTML.gif
(4.1)
It is easy to see that conditions (i), (ii), and (iii) of Theorem 3.1 hold. Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ56_HTML.gif
(4.2)

then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq364_HTML.gif . Applying Theorem 3.1, then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq365_HTML.gif , when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq366_HTML.gif , problem (4.1) has at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq367_HTML.gif distinct pairs of nontrivial classical solutions.

Example 4.2.

Consider boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ57_HTML.gif
(4.3)
It is easy to see that conditions (i), (ii), and (iii) of Theorem 3.3 hold. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq368_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ58_HTML.gif
(4.4)

then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq369_HTML.gif . Applying Theorem 3.3, then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq370_HTML.gif , when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq371_HTML.gif , problem (4.3) has at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq372_HTML.gif distinct pairs of nontrivial classical solutions.

Example 4.3.

Consider boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ59_HTML.gif
(4.5)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq373_HTML.gif , it is easy to see that conditions (i), (ii), and (iii) of Theorem 3.5 hold. Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ60_HTML.gif
(4.6)

then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq374_HTML.gif . Applying Theorem 3.5, then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq375_HTML.gif , when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq376_HTML.gif , problem (4.5) has at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq377_HTML.gif distinct pairs of nontrivial classical solutions.

Example 4.4.

Consider boundary value problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ61_HTML.gif
(4.7)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq378_HTML.gif , it is easy to see that conditions (i), (ii), and (iii) of Theorem 3.7 hold. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq379_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_Equ62_HTML.gif
(4.8)

then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq380_HTML.gif . Applying Theorem 3.7, then for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq381_HTML.gif , when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq382_HTML.gif , problem (4.7) has at least https://static-content.springer.com/image/art%3A10.1155%2F2010%2F325415/MediaObjects/13661_2010_Article_916_IEq383_HTML.gif distinct pairs of nontrivial classical solutions.

Declarations

Acknowledgments

This work was supported by the NNSF of China (no. 10871062) and a project supported by Hunan Provincial Natural Science Foundation of China (no. 10JJ6002).

Authors’ Affiliations

(1)
Department of Mathematics, Hunan Normal University

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Copyright

© Huiwen Chen and Jianli Li. 2010

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