Nontrivial Solutions of the Asymmetric Beam System with Jumping Nonlinear Terms

  • Tacksun Jung1 and

    Affiliated with

    • Q-Heung Choi2Email author

      Affiliated with

      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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq1_HTML.gif for perturbations http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq2_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq4_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq6_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq7_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq8_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq9_HTML.gif are nonzero constants. Here http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq10_HTML.gif is the beam operator in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq11_HTML.gif , and the nonlinearity http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq13_HTML.gif be the beam operator in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq14_HTML.gif , http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq16_HTML.gif for perturbations http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ1_HTML.gif
      (1.1)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq18_HTML.gif and the nonlinearity http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq20_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ2_HTML.gif
      (1.2)

      where the nonlinearity http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq22_HTML.gif The constant http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq25_HTML.gif
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq26_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq28_HTML.gif for a perturbation http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq29_HTML.gif of the asymmetric beam equation
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ4_HTML.gif
      (1.4)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq30_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq32_HTML.gif and periodic condition on the variable http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq35_HTML.gif for perturbations http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq37_HTML.gif http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq39_HTML.gif be the differential operator and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq40_HTML.gif Then the eigenvalue problem
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ5_HTML.gif
      (2.1)
      has infinitely many eigenvalues http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq41_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq42_HTML.gif and corresponding normalized eigenfunctions http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq43_HTML.gif given by
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq44_HTML.gif are given by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ7_HTML.gif
      (2.3)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq45_HTML.gif be the square http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq46_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq47_HTML.gif the Hilbert space defined by
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq48_HTML.gif is an orthonormal basis in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq49_HTML.gif . Let us denote an element http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq50_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq51_HTML.gif as
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq52_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq53_HTML.gif as
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ11_HTML.gif
      (2.7)

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

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

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

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

      Theorem 2.2.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq67_HTML.gif , and http://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,
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq69_HTML.gif .

      Proof.

      Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq70_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq71_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq72_HTML.gif . The equation is equivalent to
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ14_HTML.gif
      (2.10)
      By Lemma 2.1, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq73_HTML.gif is a compact linear map from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq74_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq75_HTML.gif . Therefore, it is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq76_HTML.gif norm http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq77_HTML.gif . We note that
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq78_HTML.gif into http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq79_HTML.gif with Lipschitz constant http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq81_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq82_HTML.gif is a solution of (2.10), http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq86_HTML.gif for which http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq87_HTML.gif is bounded and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq88_HTML.gif as http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq90_HTML.gif be a real Banach space and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq91_HTML.gif . Suppose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq92_HTML.gif satisfies Palais-Smale condition. Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq94_HTML.gif of the critical points of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq95_HTML.gif at a given level http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq96_HTML.gif . Then there exists http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq98_HTML.gif such that we denote by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq99_HTML.gif the set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq100_HTML.gif :

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

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

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

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

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

      Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq113_HTML.gif possesses a critical value http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq114_HTML.gif . Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq115_HTML.gif can be characterized as
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ16_HTML.gif
      (3.1)
      where
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq116_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq117_HTML.gif . We define a functional on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq118_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ18_HTML.gif
      (4.1)

      Then the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq119_HTML.gif is well defined in http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq122_HTML.gif .

      Lemma 4.1 (cf. [7]).

      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq124_HTML.gif with
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq125_HTML.gif be the two-dimensional subspace of http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq127_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq128_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq129_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq130_HTML.gif be the orthogonal complement of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq131_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq132_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq133_HTML.gif denote http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq134_HTML.gif onto http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq135_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq136_HTML.gif denote http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq137_HTML.gif onto http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq138_HTML.gif . Then every element http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq139_HTML.gif is expressed by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ20_HTML.gif
      (4.3)

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

      Lemma 4.2.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq142_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq143_HTML.gif . Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq145_HTML.gif of equation
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ21_HTML.gif
      (4.4)

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

      Proof.

      Choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq151_HTML.gif and let http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ22_HTML.gif
      (4.5)
      Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq154_HTML.gif into itself, the eigenvalues of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq155_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq156_HTML.gif are http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq157_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq158_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq159_HTML.gif . Therefore, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq160_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq161_HTML.gif . Since
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq163_HTML.gif (also http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq164_HTML.gif ) for fixed http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq165_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq166_HTML.gif is bounded from http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq167_HTML.gif to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq168_HTML.gif there exists a unique solution http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq169_HTML.gif of (4.4) for given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq170_HTML.gif .

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

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

      Lemma 4.3.

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

      If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq186_HTML.gif is a critical point of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq187_HTML.gif , then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq189_HTML.gif and set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq190_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq191_HTML.gif , then from (4.4)
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ29_HTML.gif
      (4.12)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq192_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ30_HTML.gif
      (4.13)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq193_HTML.gif be the two subspaces of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq194_HTML.gif defined as follows:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ31_HTML.gif
      (4.14)
      Given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq195_HTML.gif and considering the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq196_HTML.gif : http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq197_HTML.gif defined by
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ32_HTML.gif
      (4.15)
      the function http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq198_HTML.gif has continuous partial Fréchet derivatives http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq199_HTML.gif and http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ33_HTML.gif
      (4.16)
      Therefore, let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq201_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq202_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq203_HTML.gif . Then by Lemma 4.2
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ34_HTML.gif
      (4.17)
      If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq204_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq205_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ35_HTML.gif
      (4.18)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq206_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq207_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq208_HTML.gif , it is easy to know that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ36_HTML.gif
      (4.19)
      And
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ37_HTML.gif
      (4.20)
      It follows that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ38_HTML.gif
      (4.21)

      Therefore, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq210_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq211_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq212_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq213_HTML.gif are in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq214_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq215_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ39_HTML.gif
      (4.22)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq216_HTML.gif . Therefore, http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq218_HTML.gif .

      Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq220_HTML.gif ] implies that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq221_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq222_HTML.gif with respect to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq223_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ41_HTML.gif
      (4.24)

      Suppose that there exists http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq224_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq225_HTML.gif . From (4.24), it follows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq226_HTML.gif for all http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq228_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq229_HTML.gif . Therefore, http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq232_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq233_HTML.gif for any http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq234_HTML.gif .

      Lemma 4.4.

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

      Proof.

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

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq271_HTML.gif . It follows that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq273_HTML.gif .

      Proposition 4.5.

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

      Proof.

      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq278_HTML.gif is invariant under http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq279_HTML.gif and under the map http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq280_HTML.gif . So the spectrum http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq281_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq282_HTML.gif restricted to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq283_HTML.gif contains http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq284_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq285_HTML.gif . The spectrum http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq286_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq287_HTML.gif restricted to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq288_HTML.gif contains http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq289_HTML.gif in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq291_HTML.gif of this equation satisfies http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq294_HTML.gif

      Lemma 4.6.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq295_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq296_HTML.gif . Then the functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq297_HTML.gif , defined on http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq300_HTML.gif is bounded and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq301_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq302_HTML.gif . Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq304_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq305_HTML.gif .

      Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq307_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq308_HTML.gif . So
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq309_HTML.gif , also http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq310_HTML.gif . Dividing by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq311_HTML.gif and taking http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq312_HTML.gif , we get
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ45_HTML.gif
      (4.28)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq313_HTML.gif , we get http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq314_HTML.gif weakly in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq315_HTML.gif . Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq317_HTML.gif strongly in http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ46_HTML.gif
      (4.29)
      with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq320_HTML.gif the following equation
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq321_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq322_HTML.gif .

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

      The above equation ( http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq327_HTML.gif has only one critical point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq328_HTML.gif .

      Given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq329_HTML.gif , let http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ50_HTML.gif
      (4.33)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq331_HTML.gif . Let us define the reduced functional http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq332_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq333_HTML.gif by http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq335_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq336_HTML.gif by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq337_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq338_HTML.gif . We also note that, for http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq339_HTML.gif has only the critical point http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq340_HTML.gif .

      Lemma 4.7.

      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq341_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq342_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq343_HTML.gif . Then we have http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq344_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq345_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq347_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq348_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq349_HTML.gif . Then we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ51_HTML.gif
      (4.34)

      for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq350_HTML.gif (certainly for also the norm http://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
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq352_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq353_HTML.gif and a constant http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq354_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ53_HTML.gif
      (4.36)
      Given http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq355_HTML.gif , let http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ54_HTML.gif
      (4.37)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq357_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq358_HTML.gif . By dividing http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq359_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ55_HTML.gif
      (4.38)

      By Lemma 4.2, http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq360_HTML.gif is Lipschitz continuous on http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq361_HTML.gif . So the sequence http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq362_HTML.gif is bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq363_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq364_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq365_HTML.gif ), it follows that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq366_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq367_HTML.gif are bounded in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq368_HTML.gif . Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq370_HTML.gif converging to some http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq371_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq372_HTML.gif , denoted by itself. Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq374_HTML.gif converges to http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq375_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq377_HTML.gif converges to an element http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq378_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq379_HTML.gif .

      On the other hand, since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq380_HTML.gif , dividing this inequality by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq381_HTML.gif , we get
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq382_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ57_HTML.gif
      (4.40)
      If we set http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq383_HTML.gif in (4.40) and divide by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq384_HTML.gif , then we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ58_HTML.gif
      (4.41)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq385_HTML.gif be arbitrary. Dividing (4.40) by http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq386_HTML.gif and letting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq387_HTML.gif , we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ59_HTML.gif
      (4.42)
      where http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq389_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq390_HTML.gif . Put http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq391_HTML.gif . Letting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq392_HTML.gif in (4.41), we obtain
      http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ61_HTML.gif
      (4.44)
      Letting http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq393_HTML.gif in (4.39), we obtain
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ62_HTML.gif
      (4.45)

      Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq394_HTML.gif , this contradicts to the fact that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq395_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq396_HTML.gif . This proves that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq398_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq399_HTML.gif , and http://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
      http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq402_HTML.gif is a critical point of functional http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq405_HTML.gif of 0 in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq406_HTML.gif such that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq408_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq409_HTML.gif from Lemma 4.8 and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq411_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq412_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ64_HTML.gif
      (4.47)
      Let http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq413_HTML.gif be an open neighborhood of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq414_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq415_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq416_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq417_HTML.gif , we can choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq418_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq419_HTML.gif . Since http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ65_HTML.gif
      (4.48)

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

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

      If http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq432_HTML.gif , then define
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ66_HTML.gif
      (4.49)
      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq433_HTML.gif . Hence,
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ67_HTML.gif
      (4.50)
      That is http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq434_HTML.gif . By contradiction, assume http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq435_HTML.gif . Use the functional http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq437_HTML.gif . We choose http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq438_HTML.gif such that http://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), http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq440_HTML.gif and
      http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq441_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq442_HTML.gif at level http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq443_HTML.gif such that http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq446_HTML.gif for perturbations http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ69_HTML.gif
      (5.1)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq448_HTML.gif and the nonlinearity http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq450_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq451_HTML.gif , and http://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 http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ70_HTML.gif
      (5.2)

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

      Let http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ71_HTML.gif
      (5.3)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq456_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq457_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq459_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq460_HTML.gif . Hence we get the solutions http://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:
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ72_HTML.gif
      (5.4)

      where http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq462_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq463_HTML.gif . When http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq465_HTML.gif . When http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq467_HTML.gif , http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq470_HTML.gif , and http://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 http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ73_HTML.gif
      (5.5)

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

      Let http://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
      http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_Equ74_HTML.gif
      (5.6)
      Since http://static-content.springer.com/image/art%3A10.1155%2F2010%2F728101/MediaObjects/13661_2009_Article_952_IEq475_HTML.gif , and http://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 http://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:
      http://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 http://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

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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.