Open Access

Nontrivial Solutions of the Asymmetric Beam System with Jumping Nonlinear Terms

Boundary Value Problems20102010:728101

DOI: 10.1155/2010/728101

Received: 8 October 2009

Accepted: 11 September 2010

Published: 21 September 2010

Abstract

We investigate the existence of multiple nontrivial solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq1_HTML.gif for perturbations https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq2_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq3_HTML.gif of the beam system with Dirichlet boundary condition https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq4_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq5_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq6_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq7_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq8_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq9_HTML.gif are nonzero constants. Here https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq10_HTML.gif is the beam operator in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq11_HTML.gif , and the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq12_HTML.gif crosses the eigenvalues of the beam operator.

1. Introduction

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq13_HTML.gif be the beam operator in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq14_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq15_HTML.gif In this paper, we investigate the existence of multiple nontrivial solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq16_HTML.gif for perturbations https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq17_HTML.gif of the beam system with Dirichlet boundary condition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq18_HTML.gif and the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq19_HTML.gif crosses the eigenvalues of the beam operator. This system represents a bending beam supported by cables in the two directions.

In [1, 2], the authors investigated the multiplicity of solutions of a nonlinear suspension bridge equation in an interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq20_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ2_HTML.gif
(1.2)

where the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq21_HTML.gif crosses an eigenvalue. This equation represents a bending beam supported by cables under a load https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq22_HTML.gif The constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq23_HTML.gif represents the restoring force if the cables stretch. The nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq24_HTML.gif models the fact that cables resist expansion but do not resist compression.

In [2] Lazer and McKenna point out that the kind of nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq25_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ3_HTML.gif
(1.3)

can furnish a model to study travelling waves in suspension bridges. This is a one-dimensional beam equation that represents only the up and down travelling waves of the beam. But the beam has also the right and left travelling waves. Hence we can consider two-dimensional beam equation (1.1).

The nonlinear equation with jumping nonlinearity has been extensively studied by many authors. For the fourth order elliptic equation, Taratello [3] and Micheletti and Pistoia [4, 5] proved the existence of nontrivial solutions, by using degree theory and critical point theory, separately. For one-dimensional case, Lazer and McKenna [6] proved the existence of nontrivial solution by the global bifurcation method. For this jumping nonlinearity, we are interested in the multiple nontrivial solutions of the equation. Here we used variational reduction method to find the nontrivial solutions of problem (1.1).

In Section 2, we investigate some properties of the Hilbert space spanned by eigenfunctions of the beam operator. We show that only the trivial solution exists for problem (1.4) when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq26_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq27_HTML.gif . In Section 3, we state the Mountain Pass Theorem. In Section 4, we investigate the existence of nontrivial solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq28_HTML.gif for a perturbation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq29_HTML.gif of the asymmetric beam equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ4_HTML.gif
(1.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq30_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq31_HTML.gif are constants. This equation satisfies Dirichlet boundary condition on the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq32_HTML.gif and periodic condition on the variable https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq33_HTML.gif . We use the variational reduction method to apply mountain pass theorem in order to get the main result that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq34_HTML.gif (1.2) has at least three periodic solutions, two of which are nontrivial. In Section 5, we investigate the existence of multiple nontrivial solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq35_HTML.gif for perturbations https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq36_HTML.gif of beam system (1.1). We also prove that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq37_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq38_HTML.gif (1.1) has only the trivial solution.

2. Preliminaries

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq39_HTML.gif be the differential operator and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq40_HTML.gif Then the eigenvalue problem
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ5_HTML.gif
(2.1)
has infinitely many eigenvalues https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq41_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq42_HTML.gif and corresponding normalized eigenfunctions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq43_HTML.gif given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ6_HTML.gif
(2.2)
We note that all eigenvalues in the interval https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq44_HTML.gif are given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ7_HTML.gif
(2.3)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq45_HTML.gif be the square https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq46_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq47_HTML.gif the Hilbert space defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ8_HTML.gif
(2.4)
Then the set of functions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq48_HTML.gif is an orthonormal basis in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq49_HTML.gif . Let us denote an element https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq50_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq51_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ9_HTML.gif
(2.5)
and we define a subspace https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq52_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq53_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ10_HTML.gif
(2.6)
Then this is a complete normed space with a norm
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ11_HTML.gif
(2.7)

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq54_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq55_HTML.gif , we have that

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq57_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq58_HTML.gif denotes the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq59_HTML.gif norm of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq60_HTML.gif ;

() https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq62_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq63_HTML.gif .

Define https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq64_HTML.gif . Then we have the following lemma (cf. [7]).

Lemma 2.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq65_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq66_HTML.gif . Then we have that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ12_HTML.gif
(2.8)

Theorem 2.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq67_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq68_HTML.gif . Then the equation, with Dirichlet boundary condition,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ13_HTML.gif
(2.9)

has only the trivial solution in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq69_HTML.gif .

Proof.

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq70_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq71_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq72_HTML.gif . The equation is equivalent to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ14_HTML.gif
(2.10)
By Lemma 2.1, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq73_HTML.gif is a compact linear map from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq74_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq75_HTML.gif . Therefore, it is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq76_HTML.gif norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq77_HTML.gif . We note that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ15_HTML.gif
(2.11)

So the right-hand side of (2.10) defines a Lipschitz mapping of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq78_HTML.gif into https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq79_HTML.gif with Lipschitz constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq80_HTML.gif . Therefore, by the contraction mapping principle, there exists a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq81_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq82_HTML.gif is a solution of (2.10), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq83_HTML.gif is the unique solution.

3. Mountain Pass Theorem

The mountain pass theorem concerns itself with proving the existence of critical points of functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq84_HTML.gif which satisfy the Palais-Smale (PS) condition, which occurs repeatedly in critical point theory.

Definition 3.1.

We say that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq85_HTML.gif satisfies the Palais-Smale condition if any sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq86_HTML.gif for which https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq87_HTML.gif is bounded and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq88_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq89_HTML.gif possesses a convergent sequence.

The following deformation theorem is stated in [8].

Theorem 3.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq90_HTML.gif be a real Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq91_HTML.gif . Suppose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq92_HTML.gif satisfies Palais-Smale condition. Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq93_HTML.gif be a given neighborhood of the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq94_HTML.gif of the critical points of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq95_HTML.gif at a given level https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq96_HTML.gif . Then there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq97_HTML.gif , as small as we want, and a deformation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq98_HTML.gif such that we denote by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq99_HTML.gif the set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq100_HTML.gif :

(i) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq101_HTML.gif ,

(ii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq102_HTML.gif ,

(iii) https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq103_HTML.gif .

We state the Mountain Pass Theorem.

Theorem 3.3.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq104_HTML.gif be a real Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq105_HTML.gif satisfy https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq106_HTML.gif condition. Suppose that

() there are constants https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq108_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq109_HTML.gif , and

() there is an https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq111_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq112_HTML.gif .

Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq113_HTML.gif possesses a critical value https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq114_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq115_HTML.gif can be characterized as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ16_HTML.gif
(3.1)
where
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ17_HTML.gif
(3.2)

4. Critical Point Theory and Multiple Nontrivial Solutions

We investigate the existence of multiple solutions of (1.1) when https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq116_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq117_HTML.gif . We define a functional on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq118_HTML.gif by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ18_HTML.gif
(4.1)

Then the functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq119_HTML.gif is well defined in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq120_HTML.gif and the solutions of (1.4) coincide with the critical points of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq121_HTML.gif . Now we investigate the property of functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq122_HTML.gif .

Lemma 4.1 (cf. [7]).

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq123_HTML.gif is continuous and Frechet differentiable at each https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq124_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ19_HTML.gif
(4.2)

We will use a variational reduction method to apply the mountain pass theorem.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq125_HTML.gif be the two-dimensional subspace of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq126_HTML.gif . Both of them have the same eigenvalue https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq127_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq128_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq129_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq130_HTML.gif be the orthogonal complement of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq131_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq132_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq133_HTML.gif denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq134_HTML.gif onto https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq135_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq136_HTML.gif denote https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq137_HTML.gif onto https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq138_HTML.gif . Then every element https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq139_HTML.gif is expressed by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ20_HTML.gif
(4.3)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq140_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq141_HTML.gif .

Lemma 4.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq142_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq143_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq144_HTML.gif be given. Then we have that there exists a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq145_HTML.gif of equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ21_HTML.gif
(4.4)

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq146_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq147_HTML.gif satisfies a uniform Lipschitz continuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq148_HTML.gif with respect to the https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq149_HTML.gif norm (also the norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq150_HTML.gif ).

Proof.

Choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq151_HTML.gif and let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq152_HTML.gif Then (4.4) can be written as
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ22_HTML.gif
(4.5)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq153_HTML.gif is a self-adjoint, compact, linear map from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq154_HTML.gif into itself, the eigenvalues of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq155_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq156_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq157_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq158_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq159_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq160_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq161_HTML.gif . Since
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ23_HTML.gif
(4.6)

the right-hand side of (4.5) defines a Lipschitz mapping because for fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq162_HTML.gif maps into itself. By the contraction mapping principle, there exists a unique https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq163_HTML.gif (also https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq164_HTML.gif ) for fixed https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq165_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq166_HTML.gif is bounded from https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq167_HTML.gif to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq168_HTML.gif there exists a unique solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq169_HTML.gif of (4.4) for given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq170_HTML.gif .

Let
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ24_HTML.gif
(4.7)
Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq171_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq172_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq173_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq174_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq175_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ25_HTML.gif
(4.8)
Hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ26_HTML.gif
(4.9)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq176_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ27_HTML.gif
(4.10)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq177_HTML.gif is continuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq178_HTML.gif with respect to norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq179_HTML.gif (also, to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq180_HTML.gif ).

Lemma 4.3.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq181_HTML.gif is defined by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq182_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq183_HTML.gif is a continuous Frechet derivative https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq184_HTML.gif with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq185_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ28_HTML.gif
(4.11)

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq186_HTML.gif is a critical point of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq187_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq188_HTML.gif is a solution of (1.4) and conversely every solution of (1.4) is of this form.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq189_HTML.gif and set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq190_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq191_HTML.gif , then from (4.4)
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ29_HTML.gif
(4.12)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq192_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ30_HTML.gif
(4.13)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq193_HTML.gif be the two subspaces of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq194_HTML.gif defined as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ31_HTML.gif
(4.14)
Given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq195_HTML.gif and considering the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq196_HTML.gif : https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq197_HTML.gif defined by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ32_HTML.gif
(4.15)
the function https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq198_HTML.gif has continuous partial Fréchet derivatives https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq199_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq200_HTML.gif with respect to its first and second variables given by
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ33_HTML.gif
(4.16)
Therefore, let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq201_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq202_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq203_HTML.gif . Then by Lemma 4.2
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ34_HTML.gif
(4.17)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq204_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq205_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ35_HTML.gif
(4.18)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq206_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq207_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq208_HTML.gif , it is easy to know that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ36_HTML.gif
(4.19)
And
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ37_HTML.gif
(4.20)
It follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ38_HTML.gif
(4.21)

Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq209_HTML.gif is strictly convex with respect to the second variable.

Similarly, using the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq210_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq211_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq212_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq213_HTML.gif are in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq214_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq215_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ39_HTML.gif
(4.22)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq216_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq217_HTML.gif is strictly concave with respect to the first variable. From (4.17), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ40_HTML.gif
(4.23)

with equality if and only if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq218_HTML.gif .

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq219_HTML.gif is strictly concave (convex) with respect to its first (second) variable, [9, Theorem https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq220_HTML.gif ] implies that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq221_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq222_HTML.gif with respect to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq223_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ41_HTML.gif
(4.24)

Suppose that there exists https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq224_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq225_HTML.gif . From (4.24), it follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq226_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq227_HTML.gif . Then by Lemma 4.2, it follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq228_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq229_HTML.gif . Therefore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq230_HTML.gif is a solution of (1.4).

Conversely, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq231_HTML.gif is a solution of (1.4) and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq232_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq233_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq234_HTML.gif .

Lemma 4.4.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq235_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq236_HTML.gif . Then there exists a small open neighborhood https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq237_HTML.gif of 0 in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq238_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq239_HTML.gif is a strict local minimum of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq240_HTML.gif .

Proof.

For https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq241_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq242_HTML.gif , problem (1.4) has a trivial solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq243_HTML.gif . Thus we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq244_HTML.gif . Since the subspace https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq245_HTML.gif is orthogonal complement of subspace https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq246_HTML.gif , we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq247_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq248_HTML.gif . Furthermore, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq249_HTML.gif is the unique solution of (4.4) in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq250_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq251_HTML.gif . The trivial solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq252_HTML.gif is of the form https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq253_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq254_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq255_HTML.gif is an identity map on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq256_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq257_HTML.gif is continuous, it follows that there exists a small open neighborhood https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq258_HTML.gif of 0 in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq259_HTML.gif such that if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq260_HTML.gif then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq261_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq262_HTML.gif . By Lemma 4.2, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq263_HTML.gif is the solution of (4.5) for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq264_HTML.gif . Therefore, if https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq265_HTML.gif , then for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq266_HTML.gif we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq267_HTML.gif . Thus
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ42_HTML.gif
(4.25)
If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq268_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq269_HTML.gif . Therefore, in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq270_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ43_HTML.gif
(4.26)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq271_HTML.gif . It follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq272_HTML.gif is a strict local point of minimum of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq273_HTML.gif .

Proposition 4.5.

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq274_HTML.gif , then the equation https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq275_HTML.gif admits only the trivial solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq276_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq277_HTML.gif .

Proof.

https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq278_HTML.gif is invariant under https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq279_HTML.gif and under the map https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq280_HTML.gif . So the spectrum https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq281_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq282_HTML.gif restricted to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq283_HTML.gif contains https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq284_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq285_HTML.gif . The spectrum https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq286_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq287_HTML.gif restricted to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq288_HTML.gif contains https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq289_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq290_HTML.gif . From the symmetry theorem in [10], any solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq291_HTML.gif of this equation satisfies https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq292_HTML.gif . This nontrivial periodic solution is periodic with periodic https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq293_HTML.gif . This shows that there is no nontrivial solution of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq294_HTML.gif

Lemma 4.6.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq295_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq296_HTML.gif . Then the functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq297_HTML.gif , defined on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq298_HTML.gif , satisfies the Palais-Smale condition.

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq299_HTML.gif be a Palais-Smale sequence that is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq300_HTML.gif is bounded and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq301_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq302_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq303_HTML.gif is two-dimensional, it is enough to prove that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq304_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq305_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq306_HTML.gif be the solution of (1.4) with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq307_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq308_HTML.gif . So
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ44_HTML.gif
(4.27)
By contradiction, we suppose that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq309_HTML.gif , also https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq310_HTML.gif . Dividing by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq311_HTML.gif and taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq312_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ45_HTML.gif
(4.28)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq313_HTML.gif , we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq314_HTML.gif weakly in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq315_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq316_HTML.gif is a compact operator, passing to a subsequence, we get https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq317_HTML.gif strongly in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq318_HTML.gif . Taking the limit of both sides of (4.28), it follows that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ46_HTML.gif
(4.29)
with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq319_HTML.gif . This contradicts to the fact that for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq320_HTML.gif the following equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ47_HTML.gif
(4.30)

has only the trivial solution by Proposition 4.5. Hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq321_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq322_HTML.gif .

We now define the functional on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq323_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq324_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ48_HTML.gif
(4.31)
The critical points of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq325_HTML.gif coincide with solutions of the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ49_HTML.gif
(4.32)

The above equation ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq326_HTML.gif ) has only the trivial solution and hence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq327_HTML.gif has only one critical point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq328_HTML.gif .

Given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq329_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq330_HTML.gif be the unique solution of the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ50_HTML.gif
(4.33)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq331_HTML.gif . Let us define the reduced functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq332_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq333_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq334_HTML.gif . We note that we can obtain the same results as Lemmas 4.1 and 4.2 when we replace https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq335_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq336_HTML.gif by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq337_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq338_HTML.gif . We also note that, for https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq339_HTML.gif has only the critical point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq340_HTML.gif .

Lemma 4.7.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq341_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq342_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq343_HTML.gif . Then we have https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq344_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq345_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq346_HTML.gif .

The proof of the lemma can be found in [1].

Lemma 4.8.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq347_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq348_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq349_HTML.gif . Then we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ51_HTML.gif
(4.34)

for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq350_HTML.gif (certainly for also the norm https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq351_HTML.gif ).

Proof.

Suppose that it is not true that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ52_HTML.gif
(4.35)
Then there exists a sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq352_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq353_HTML.gif and a constant https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq354_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ53_HTML.gif
(4.36)
Given https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq355_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq356_HTML.gif be the unique solution of the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ54_HTML.gif
(4.37)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq357_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq358_HTML.gif . By dividing https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq359_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ55_HTML.gif
(4.38)

By Lemma 4.2, https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq360_HTML.gif is Lipschitz continuous on https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq361_HTML.gif . So the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq362_HTML.gif is bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq363_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq364_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq365_HTML.gif ), it follows that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq366_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq367_HTML.gif are bounded in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq368_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq369_HTML.gif is a compact operator, there is a subsequence of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq370_HTML.gif converging to some https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq371_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq372_HTML.gif , denoted by itself. Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq373_HTML.gif is a two-dimensional space, assume that sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq374_HTML.gif converges to https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq375_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq376_HTML.gif . Therefore, we can get that the sequence https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq377_HTML.gif converges to an element https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq378_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq379_HTML.gif .

On the other hand, since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq380_HTML.gif , dividing this inequality by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq381_HTML.gif , we get
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ56_HTML.gif
(4.39)
By Lemma 4.2, it follows that for any https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq382_HTML.gif
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ57_HTML.gif
(4.40)
If we set https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq383_HTML.gif in (4.40) and divide by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq384_HTML.gif , then we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ58_HTML.gif
(4.41)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq385_HTML.gif be arbitrary. Dividing (4.40) by https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq386_HTML.gif and letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq387_HTML.gif , we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ59_HTML.gif
(4.42)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq388_HTML.gif Then (4.42) can be written in the form https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq389_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq390_HTML.gif . Put https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq391_HTML.gif . Letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq392_HTML.gif in (4.41), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ60_HTML.gif
(4.43)
where we have used (4.42). Hence
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ61_HTML.gif
(4.44)
Letting https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq393_HTML.gif in (4.39), we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ62_HTML.gif
(4.45)

Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq394_HTML.gif , this contradicts to the fact that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq395_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq396_HTML.gif . This proves that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq397_HTML.gif .

Now we state the main result in this paper.

Theorem 4.9.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq398_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq399_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq400_HTML.gif . Then there exist at least three solutions of the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ63_HTML.gif
(4.46)

two of which are nontrivial solutions.

Proof.

We remark that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq401_HTML.gif is the trivial solution of problem (1.4). Then https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq402_HTML.gif is a critical point of functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq403_HTML.gif . Next we want to find others critical points of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq404_HTML.gif which are corresponding to the solutions of problem (1.4).

By Lemma 4.4, there exists a small open neighborhood https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq405_HTML.gif of 0 in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq406_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq407_HTML.gif is a strict local point of minimum of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq408_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq409_HTML.gif from Lemma 4.8 and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq410_HTML.gif is a two-dimensional space, there exists a critical point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq411_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq412_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ64_HTML.gif
(4.47)
Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq413_HTML.gif be an open neighborhood of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq414_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq415_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq416_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq417_HTML.gif , we can choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq418_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq419_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq420_HTML.gif satisfies the Palais-Smale condition, by the Mountain Pass Theorem (Theorem 3.3), there is a critical value
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ65_HTML.gif
(4.48)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq421_HTML.gif

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq422_HTML.gif , then there exists a critical point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq423_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq424_HTML.gif at level https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq425_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq426_HTML.gif , 0 ( since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq427_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq428_HTML.gif ). Therefore, in case https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq429_HTML.gif , the functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq430_HTML.gif has also at least 3 critical points https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq431_HTML.gif .

If https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq432_HTML.gif , then define
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ66_HTML.gif
(4.49)
where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq433_HTML.gif . Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ67_HTML.gif
(4.50)
That is https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq434_HTML.gif . By contradiction, assume https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq435_HTML.gif . Use the functional https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq436_HTML.gif for the deformation theorem (Theorem 4.9) and taking https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq437_HTML.gif . We choose https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq438_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq439_HTML.gif . From the deformation theorem (Theorem 3.2), https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq440_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ68_HTML.gif
(4.51)

which is a contradiction. Therefore, there exists a critical point https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq441_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq442_HTML.gif at level https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq443_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq444_HTML.gif , 0, which means that (1.4) has at least three critical points. Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq445_HTML.gif , these two critical points coincide with two nontrivial period solutions of problem (1.4).

5. Nontrivial Solutions for the Beam System

In this section, we investigate the existence of multiple nontrivial solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq446_HTML.gif for perturbations https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq447_HTML.gif of the beam system with Dirichlet boundary condition
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ69_HTML.gif
(5.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq448_HTML.gif and the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq449_HTML.gif crosses the eigenvalues of the beam operator.

Theorem 5.1.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq450_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq451_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq452_HTML.gif . Then beam system (5.1) has at least three solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq453_HTML.gif , two of which are nontrivial solutions.

Proof.

From problem (5.1), we get the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ70_HTML.gif
(5.2)

where the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq454_HTML.gif

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq455_HTML.gif . Then the above equation is equivalent to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ71_HTML.gif
(5.3)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq456_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq457_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq458_HTML.gif , the above equation has at least three solutions, two of which are nontrivial solutions, say https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq459_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq460_HTML.gif . Hence we get the solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq461_HTML.gif of problem (5.1) from the following systems:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ72_HTML.gif
(5.4)

where https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq462_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq463_HTML.gif . When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq464_HTML.gif , from the above equation, we get the trivial solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq465_HTML.gif . When https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq466_HTML.gif , from the above equation, we get the nontrivial solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq467_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq468_HTML.gif .Therefore, system(5.1) has at least three solutions https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq469_HTML.gif , two of which are nontrivial solutions.

Theorem 5.2.

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq470_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq471_HTML.gif . Then system (5.1) has only the trivial solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq472_HTML.gif .

Proof.

From problem (5.1), we get the equation
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ73_HTML.gif
(5.5)

where the nonlinearity https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq473_HTML.gif

Let https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq474_HTML.gif . Then the above equation is equivalent to
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ74_HTML.gif
(5.6)
Since https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq475_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq476_HTML.gif , by Theorem 2.2, the above equation has the trivial solution. Hence we have the trivial solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq477_HTML.gif of problem (5.1) from the following system:
https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ75_HTML.gif
(5.7)

From (5.9), we get the trivial solution https://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq478_HTML.gif .

Declarations

Acknowledgments

This work (Choi) was supported by Inha University Research Grant. The authors appreciate very much the referee's corrections and revisions.

Authors’ Affiliations

(1)
Department of Mathematics, Kunsan National University
(2)
Department of Mathematics Education, Inha University

References

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Copyright

© The Author(s) Tacksun Jung and Q-Heung Choi. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.