Existence of Four Solutions of Some Nonlinear Hamiltonian System

  • Tacksun Jung1 and

    Affiliated with

    • Q-Heung Choi2Email author

      Affiliated with

      Boundary Value Problems20082008:293987

      DOI: 10.1155/BVP/2008/293987

      Received: 25 August 2007

      Accepted: 3 December 2007

      Published: 14 January 2008

      Abstract

      We show the existence of four http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq1_HTML.gif -periodic solutions of the nonlinear Hamiltonian system with some conditions. We prove this problem by investigating the geometry of the sublevels of the functional and two pairs of sphere-torus variational linking inequalities of the functional and applying the critical point theory induced from the limit relative category.

      1. Introduction and Statements of Main Results

      Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq2_HTML.gif be a http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq3_HTML.gif function defined on http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq4_HTML.gif which is http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq5_HTML.gif -periodic with respect to the first variable http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq6_HTML.gif .In this paper, we investigate the number of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq7_HTML.gif -periodic nontrivial solutions of the following nonlinear Hamiltonian system
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ1_HTML.gif
      (1.1)
      where http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq8_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq9_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ2_HTML.gif
      (1.2)
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq10_HTML.gif is the identity matrix on http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq11_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq12_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq13_HTML.gif is the gradient of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq14_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq15_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq16_HTML.gif Then (1.1) can be rewritten as
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ3_HTML.gif
      (1.3)

      We assume that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq17_HTML.gif satisfies the following conditions.

      (H1)There exist constants http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq18_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ4_HTML.gif
      (1.4)
      (H2)Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq19_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq20_HTML.gif be integers and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq21_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq22_HTML.gif be any numbers (without loss of generality, we may assume http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq23_HTML.gif ) such that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq24_HTML.gif . Suppose that there exist http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq25_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq26_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq27_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ5_HTML.gif
      (1.5)
      (H 3) http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq28_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq29_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq30_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ6_HTML.gif
      (1.6)

      (H4) http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq31_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq32_HTML.gif -periodic with respect to http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq33_HTML.gif .

      We are looking for the weak solutions of (1.1). Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq34_HTML.gif . The http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq35_HTML.gif -periodic weak solution http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq36_HTML.gif of (1.3) satisfies
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ7_HTML.gif
      (1.7)
      and coincides with the critical points of the induced functional
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ8_HTML.gif
      (1.8)

      where http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq37_HTML.gif .

      Our main results are the following.

      Theorem 1.1.

      Assume that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq38_HTML.gif satisfies conditions http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq39_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq40_HTML.gif . Then there exists a number http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq41_HTML.gif such that for any http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq42_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq43_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq44_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq45_HTML.gif , system (1.1) has at least four nontrivial http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq46_HTML.gif -periodic solutions.

      Theorem 1.2.

      Assume that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq47_HTML.gif satisfies conditions http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq48_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq49_HTML.gif . Then there exists a number http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq50_HTML.gif such that for any http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq51_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq52_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq53_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq54_HTML.gif , system (1.1) has at least four nontrivial http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq55_HTML.gif -periodic solutions.

      Chang proved in [1] that, under conditions http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq56_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq57_HTML.gif ,system (1.1) has at least two nontrivial http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq58_HTML.gif -periodic solutions. He proved this result by using the finite dimensional variational reduction method. He first investigate the critical points of the functional on the finite dimensional subspace and the http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq59_HTML.gif condition of the reduced functional and find one critical point of the mountain pass type. He also found another critical point by the shape of graph of the reduced functional.

      For the proofs of Theorems 1.1 and ,1.2 we first separate the whole space http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq60_HTML.gif into the four mutually disjoint four subspaces http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq61_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq62_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq63_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq64_HTML.gif which are introduced in Section 3 and then we investigate two pairs of sphere-torus variational linking inequalities of the reduced functional http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq65_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq66_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq67_HTML.gif on the submanifold with boundary http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq68_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq69_HTML.gif , respectively, and translate these two pairs of sphere-torus variational links of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq70_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq71_HTML.gif into the two pairs of torus-sphere variational links of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq72_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq73_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq74_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq75_HTML.gif are the restricted functionals of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq76_HTML.gif to the manifold with boundary http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq77_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq78_HTML.gif , respectively. Since http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq79_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq80_HTML.gif are strongly indefinite functinals, we use the notion of the http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq81_HTML.gif condition and the limit relative category instead of the notion of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq82_HTML.gif condition and the relative category, which are the useful tools for the proofs of the main theorems. We also investigate the limit relative category of torus in (torus, boundary of torus) on http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq83_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq84_HTML.gif , respectively. By the critical point theory induced from the limit relative category theory we obtain two nontrivial http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq85_HTML.gif -periodic solutions in each subspace http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq86_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq87_HTML.gif , so we obtain at least four nontrivial http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq88_HTML.gif -periodic solutions of (1.1).

      In Section 2, we introduce some notations and some notions of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq89_HTML.gif condition and the limit relative category and recall the critical point theory on the manifold with boundary. We also prove some propositions. In Section 3, we prove Theorem 1.1 and in Section 4, we prove Theorem 1.2.

      2. Recall of the Critical Point Theory Induced from the Limit Relative Category

      Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq90_HTML.gif The scalar product in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq91_HTML.gif naturally extends as the duality pairing between http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq92_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq93_HTML.gif . It is known that if http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq94_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq95_HTML.gif -periodic, then it has a Fourier expansion http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq96_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq97_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq98_HTML.gif : http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq99_HTML.gif is the closure of such functions with respect to the norm
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ9_HTML.gif
      (2.1)
      Let us set the functional
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ10_HTML.gif
      (2.2)
      ,
      so that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ11_HTML.gif
      (2.3)
      Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq100_HTML.gif denote the usual bases in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq101_HTML.gif and set
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ12_HTML.gif
      (2.4)
      Then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq102_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq103_HTML.gif are the subspaces of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq104_HTML.gif on which http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq105_HTML.gif is null, positive definite and negative definite, and these spaces are orthogonal with respect to the bilinear form
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ13_HTML.gif
      (2.5)
      associated with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq106_HTML.gif . Here, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq107_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq108_HTML.gif If http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq109_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq110_HTML.gif , then the bilinear form is zero and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq111_HTML.gif . We also note that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq112_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq113_HTML.gif are mutually orthogonal in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq114_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq115_HTML.gif be the projection from http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq116_HTML.gif onto http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq117_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq118_HTML.gif the one from http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq119_HTML.gif onto http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq120_HTML.gif . Then the norm in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq121_HTML.gif is given by
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ14_HTML.gif
      (2.6)

      which is equivalent to the usual one. The space http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq122_HTML.gif with this norm is a Hilbert space.

      We need the following facts which are proved in [2].

      Proposition 2.1.

      For each http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq123_HTML.gif , E is compactly embedded in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq124_HTML.gif . In particular, there is an http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq125_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ15_HTML.gif
      (2.7)

      for all http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq126_HTML.gif .

      Proposition 2.2.

      Assume that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq127_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq128_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq129_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq130_HTML.gif is continuous and Fréchet differentiable in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq131_HTML.gif with Fréchet derivative
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ16_HTML.gif
      (2.8)

      where http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq132_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq133_HTML.gif Moreover, the functional http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq134_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq135_HTML.gif

      Proof.

      For http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq136_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ17_HTML.gif
      (2.9)
      We have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ18_HTML.gif
      (2.10)
      Thus, we have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ19_HTML.gif
      (2.11)
      Next, we prove that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq137_HTML.gif is continuous. For http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq138_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ20_HTML.gif
      (2.12)

      Similarly, it is easily checked that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq139_HTML.gif is http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq140_HTML.gif .

      Now, we consider the critical point theory on the manifold with boundary induced from the limit relative category. Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq141_HTML.gif be a Hilbert space and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq142_HTML.gif be the closure of an open subset of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq143_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq144_HTML.gif can be endowed with the structure of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq145_HTML.gif manifold with boundary. Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq146_HTML.gif be a http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq147_HTML.gif functional, where http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq148_HTML.gif is an open set containing http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq149_HTML.gif . The http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq150_HTML.gif condition and the limit relative category (see [3]) are useful tools for the proof of the main theorem.

      Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq151_HTML.gif be a sequence of a closed finite dimensional subspace of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq152_HTML.gif with the following assumptions: http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq153_HTML.gif where http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq154_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq155_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq156_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq157_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq158_HTML.gif are subspaces of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq159_HTML.gif ), http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq160_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq161_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq162_HTML.gif are dense in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq163_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq164_HTML.gif , for any http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq165_HTML.gif , be the closure of an open subset of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq166_HTML.gif and has the structure of a http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq167_HTML.gif manifold with boundary in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq168_HTML.gif . We assume that for any http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq169_HTML.gif there exists a retraction http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq170_HTML.gif . For a given http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq171_HTML.gif , we will write http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq172_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq173_HTML.gif be a closed subspace of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq174_HTML.gif .

      Definition 2.3.

      Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq175_HTML.gif be a closed subset of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq176_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq177_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq178_HTML.gif be the relative category of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq179_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq180_HTML.gif . We define the limit relative category of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq181_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq182_HTML.gif , with respect to http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq183_HTML.gif , by
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ21_HTML.gif
      (2.13)
      We set
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ22_HTML.gif
      (2.14)

      We have the following multiplicity theorem (for the proof, see [4]).

      Theorem 2.4.

      Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq184_HTML.gif and assume that

      (1) http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq185_HTML.gif ,

      (2) http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq186_HTML.gif ,

      (3)the http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq187_HTML.gif condition with respect to http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq188_HTML.gif holds.

      Then there exists a lower critical point http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq189_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq190_HTML.gif . If
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ23_HTML.gif
      (2.15)
      then
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ24_HTML.gif
      (2.16)

      Now, we state the following multiplicity result (for the proof, see [4, Theorem 4.6]) which will be used in the proofs of our main theorems.

      Theorem 2.5.

      Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq191_HTML.gif be a Hilbert space and let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq192_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq193_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq194_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq195_HTML.gif are three closed subspaces of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq196_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq197_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq198_HTML.gif of finite dimension. For a given subspace http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq199_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq200_HTML.gif , let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq201_HTML.gif be the orthogonal projection from http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq202_HTML.gif onto http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq203_HTML.gif . Set
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ25_HTML.gif
      (2.17)
      and let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq204_HTML.gif be a http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq205_HTML.gif function defined on a neighborhood http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq206_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq207_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq208_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq209_HTML.gif . One defines
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ26_HTML.gif
      (2.18)
      Assume that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ27_HTML.gif
      (2.19)
      and that the http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq210_HTML.gif condition holds for http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq211_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq212_HTML.gif , with respect to the sequrnce http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq213_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq214_HTML.gif , where
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ28_HTML.gif
      (2.20)

      Moreover, one assumes http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq215_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq216_HTML.gif has no critical points http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq217_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq218_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq219_HTML.gif . Then there exist two lower critical points http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq220_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq221_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq222_HTML.gif on http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq223_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq224_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq225_HTML.gif .

      3. Proof of Theorem 1.1

      We assume that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq226_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq227_HTML.gif denote the usual bases in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq228_HTML.gif and set
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ29_HTML.gif
      (3.1)
      Then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq229_HTML.gif is the topological direct sum of subspaces http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq230_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq231_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq232_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq233_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq234_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq235_HTML.gif are finite dimensional subspaces. We also set
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ30_HTML.gif
      (3.2)

      We have the following two pairs of the sphere-torus variational linking inequalities.

      Lemma 3.1. (First Sphere-Torus Variational Linking).

      Assume that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq236_HTML.gif satisfies the conditions http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq237_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq238_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq239_HTML.gif , and the condition

      (H 2)' http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq240_HTML.gif suppose that there exist http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq241_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq242_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq243_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ31_HTML.gif
      (3.3)
      Then there exist http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq244_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq245_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq246_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq247_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq248_HTML.gif , and for any http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq249_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq250_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq251_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq252_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ32_HTML.gif
      (3.4)

      Proof.

      Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq253_HTML.gif . By http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq254_HTML.gif we have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ33_HTML.gif
      (3.5)
      for some http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq255_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq256_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq257_HTML.gif such that if http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq258_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq259_HTML.gif . Thus, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq260_HTML.gif . Moreover, if http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq261_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq262_HTML.gif , so we have http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq263_HTML.gif . Next, we will show that there exist http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq264_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq265_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq266_HTML.gif such that if http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq267_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq268_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq269_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq270_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq271_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq272_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq273_HTML.gif is a small number. Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq274_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq275_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq276_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq277_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq278_HTML.gif . By http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq279_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq280_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ34_HTML.gif
      (3.6)

      Since http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq281_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq282_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq283_HTML.gif , there exist a small number http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq284_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq285_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq286_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq287_HTML.gif such that if http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq288_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq289_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq290_HTML.gif . Thus, we have http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq291_HTML.gif . Moreover, if http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq292_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq293_HTML.gif , then we have http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq294_HTML.gif . Thus, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq295_HTML.gif . Thus, we prove the lemma.

      Lemma 3.2.

      Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq296_HTML.gif be the number introduced in Lemma 3.1. Then for any http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq297_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq298_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq299_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq300_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq301_HTML.gif is a critical point for http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq302_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq303_HTML.gif .

      Proof.

      We notice that from Lemma 3.1, for fixed http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq304_HTML.gif , the functional http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq305_HTML.gif is weakly convex in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq306_HTML.gif , while, for fixed http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq307_HTML.gif , the functional http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq308_HTML.gif is strictly concave in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq309_HTML.gif . Moreover, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq310_HTML.gif is the critical point in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq311_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq312_HTML.gif . So if http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq313_HTML.gif is another critical point for http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq314_HTML.gif , then we have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ35_HTML.gif
      (3.7)

      So we have http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq315_HTML.gif .

      Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq316_HTML.gif be the orthogonal projection from http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq317_HTML.gif onto http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq318_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ36_HTML.gif
      (3.8)
      Then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq319_HTML.gif is the smooth manifold with boundary. Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq320_HTML.gif . Let us define a functional http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq321_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ37_HTML.gif
      (3.9)
      We have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ38_HTML.gif
      (3.10)
      Let us define the functional http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq322_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ39_HTML.gif
      (3.11)
      Then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq323_HTML.gif . We note that if http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq324_HTML.gif is the critical point of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq325_HTML.gif and lies in the interior of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq326_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq327_HTML.gif is the critical point of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq328_HTML.gif . We also note that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ40_HTML.gif
      (3.12)
      Let us set
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ41_HTML.gif
      (3.13)

      We note that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq329_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq330_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq331_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq332_HTML.gif have the same topological structure as http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq333_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq334_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq335_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq336_HTML.gif , respectively.

      Lemma 3.3.

      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq337_HTML.gif satisfies the http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq338_HTML.gif condition with respect to http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq339_HTML.gif for every real number http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq340_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ42_HTML.gif
      (3.14)

      Proof.

      Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq341_HTML.gif be a sequence such that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq342_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq343_HTML.gif be a sequence in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq344_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq345_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq346_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq347_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq348_HTML.gif . Set http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq349_HTML.gif (and hence http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq350_HTML.gif ) and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq351_HTML.gif . We first consider the case in which http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq352_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq353_HTML.gif . Since for http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq354_HTML.gif large http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq355_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ43_HTML.gif
      (3.15)
      By (3.9) and (3.10),
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ44_HTML.gif
      (3.16)
      In the first case, the claim follows from the limit Palais-Smale condition for http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq356_HTML.gif . In the second case, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq357_HTML.gif . We claim that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq358_HTML.gif is bounded. By contradiction, we suppose that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq359_HTML.gif and set http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq360_HTML.gif . Up to a subsequence http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq361_HTML.gif weakly for some http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq362_HTML.gif . By the asymptotically linearity of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq363_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ45_HTML.gif
      (3.17)
      We have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ46_HTML.gif
      (3.18)
      where http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq364_HTML.gif . Passing to the limit we, get
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ47_HTML.gif
      (3.19)
      Since http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq365_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq366_HTML.gif are bounded and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq367_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq368_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq369_HTML.gif . On the other hand, we have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ48_HTML.gif
      (3.20)
      Moreover, we have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ49_HTML.gif
      (3.21)
      Since http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq370_HTML.gif converges to 0 weakly and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq371_HTML.gif is bounded, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq372_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq373_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq374_HTML.gif converges to 0 strongly, which is a contradiction. Hence, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq375_HTML.gif is bounded. Up to a subsequence, we can suppose that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq376_HTML.gif converges to http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq377_HTML.gif for some http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq378_HTML.gif . We claim that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq379_HTML.gif converges to http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq380_HTML.gif strongly. We have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ50_HTML.gif
      (3.22)
      By http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq381_HTML.gif and the boundedness of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq382_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ51_HTML.gif
      (3.23)
      That is, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq383_HTML.gif converges. Since http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq384_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq385_HTML.gif converges, so http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq386_HTML.gif converges to http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq387_HTML.gif strongly. Therefore, we have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ52_HTML.gif
      (3.24)

      So we proved the first case.

      We consider the case http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq388_HTML.gif , that is, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq389_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq390_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq391_HTML.gif . In this case, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq392_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq393_HTML.gif . Thus, by the same argument as the first case, we obtain the conclusion. So we prove the lemma.

      Proposition 3.4.

      Assume that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq394_HTML.gif satisfies the conditions http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq395_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq396_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq397_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq398_HTML.gif . Then there exists a number http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq399_HTML.gif such that for any http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq400_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq401_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq402_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq403_HTML.gif , there exist at least two nontrivial critical points http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq404_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq405_HTML.gif , in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq406_HTML.gif for the functional http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq407_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ53_HTML.gif
      (3.25)

      where http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq408_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq409_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq410_HTML.gif are introduced in Lemma 3.1.

      Proof.

      First, we will find two nontrivial critical points for http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq411_HTML.gif . By Lemma 3.1, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq412_HTML.gif satisfies the torus-sphere variational linking inequality, that is, there exist http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq413_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq414_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq415_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq416_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq417_HTML.gif , and for any http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq418_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq419_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq420_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq421_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ54_HTML.gif
      (3.26)
      By Lemma 3.3, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq422_HTML.gif satisfies the http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq423_HTML.gif condition with respect to http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq424_HTML.gif for every real number http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq425_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ55_HTML.gif
      (3.27)
      Thus by Theorem 2.5, there exist two critical points http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq426_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq427_HTML.gif for the functional http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq428_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ56_HTML.gif
      (3.28)
      Setting http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq429_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq430_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ57_HTML.gif
      (3.29)
      We claim that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq431_HTML.gif , that is http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq432_HTML.gif , which implies that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq433_HTML.gif are the critical points for http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq434_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq435_HTML.gif , so http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq436_HTML.gif are the critical points for http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq437_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq438_HTML.gif . For this we assume by contradiction that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq439_HTML.gif . From (3.12), http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq440_HTML.gif , namely, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq441_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq442_HTML.gif , are the critical points for http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq443_HTML.gif . By Lemma 3.2, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq444_HTML.gif , which is a contradiction for the fact that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ58_HTML.gif
      (3.30)
      Lemma 3.2 implies that there is no critical point http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq445_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ59_HTML.gif
      (3.31)

      Hence, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq446_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq447_HTML.gif . This proves Proposition 3.4.

      Lemma 3.5. (Second Sphere-Torus Variational Linking).

      Assume that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq448_HTML.gif satisfies the conditions http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq449_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq450_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq451_HTML.gif , and the condition

      (H 2)'' http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq452_HTML.gif suppose that there exist http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq453_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq454_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq455_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ60_HTML.gif
      (3.32)
      Then there exist http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq456_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq457_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq458_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq459_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq460_HTML.gif , and for any http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq461_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq462_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq463_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq464_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ61_HTML.gif
      (3.33)

      Proof.

      Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq465_HTML.gif . By http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq466_HTML.gif we have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ62_HTML.gif
      (3.34)
      for some http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq467_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq468_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq469_HTML.gif such that if http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq470_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq471_HTML.gif . Thus we have http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq472_HTML.gif . Moreover, if http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq473_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq474_HTML.gif , so we have http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq475_HTML.gif . Next, let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq476_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq477_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq478_HTML.gif is a small number. We also let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq479_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq480_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq481_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq482_HTML.gif . By http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq483_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq484_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ63_HTML.gif
      (3.35)

      Since http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq485_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq486_HTML.gif , there exist a small number http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq487_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq488_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq489_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq490_HTML.gif such that if http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq491_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq492_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq493_HTML.gif . Thus we have http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq494_HTML.gif .

      Moreover, if http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq495_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq496_HTML.gif . Thus we have http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq497_HTML.gif . Thus we prove the lemma.

      Lemma 3.6.

      For any http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq498_HTML.gif there exists a constant http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq499_HTML.gif such that for any http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq500_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq501_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq502_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq503_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq504_HTML.gif is a critical point for http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq505_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq506_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq507_HTML.gif .

      Proof.

      By contradiction, we can suppose that there exist http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq508_HTML.gif , a sequence http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq509_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq510_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq511_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq512_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq513_HTML.gif , and a sequence http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq514_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq515_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq516_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq517_HTML.gif . We claim that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq518_HTML.gif is bounded. If we do not suppose that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq519_HTML.gif , let us set http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq520_HTML.gif . We have up to a subsequence, that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq521_HTML.gif weakly for some http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq522_HTML.gif . Furthermore,
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ64_HTML.gif
      (3.36)
      so we have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ65_HTML.gif
      (3.37)
      Moreover,
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ66_HTML.gif
      (3.38)
      so we have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ67_HTML.gif
      (3.39)
      Adding (3.37) and (3.39), we have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ68_HTML.gif
      (3.40)
      From (3.40) we have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ69_HTML.gif
      (3.41)
      We also have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ70_HTML.gif
      (3.42)
      Dividing by http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq523_HTML.gif and going to the limit, we have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ71_HTML.gif
      (3.43)
      Thus
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ72_HTML.gif
      (3.44)
      which is a contradiction since http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq524_HTML.gif . So http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq525_HTML.gif is bounded and we can suppose that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq526_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq527_HTML.gif . From (3.42), we have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ73_HTML.gif
      (3.45)
      From (3.40),
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ74_HTML.gif
      (3.46)
      Thus, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq528_HTML.gif converges to http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq529_HTML.gif strongly. We claim that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq530_HTML.gif . Assume that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq531_HTML.gif . By (H 1) http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq532_HTML.gif , for some http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq533_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq534_HTML.gif . If http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq535_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq536_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq537_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq538_HTML.gif ,
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ75_HTML.gif
      (3.47)
      If http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq539_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq540_HTML.gif , and
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ76_HTML.gif
      (3.48)
      Thus, we have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ77_HTML.gif
      (3.49)

      which is absurd because of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq541_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq542_HTML.gif . Thus http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq543_HTML.gif . We proved the lemma.

      Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq544_HTML.gif be the orthogonal projection from http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq545_HTML.gif onto http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq546_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ78_HTML.gif
      (3.50)
      Then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq547_HTML.gif is the smooth manifold with boundary. Let http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq548_HTML.gif . Let us define a functional http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq549_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ79_HTML.gif
      (3.51)
      We have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ80_HTML.gif
      (3.52)
      Let us define the functional http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq550_HTML.gif by
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ81_HTML.gif
      (3.53)
      Then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq551_HTML.gif . We note that if http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq552_HTML.gif is the critical point of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq553_HTML.gif and lies in the interior of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq554_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq555_HTML.gif is the critical point of http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq556_HTML.gif . We also note that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ82_HTML.gif
      (3.54)
      Let us set
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ83_HTML.gif
      (3.55)

      We note that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq557_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq558_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq559_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq560_HTML.gif have the same topological structure as http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq561_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq562_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq563_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq564_HTML.gif , respectively.

      We have the following lemma whose proof has the same arguments as that of Lemma 3.5 except the space http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq565_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq566_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq567_HTML.gif instead of the space http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq568_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq569_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq570_HTML.gif .

      Lemma 3.7.

      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq571_HTML.gif satisfies the http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq572_HTML.gif condition with respect to http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq573_HTML.gif for every real number http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq574_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ84_HTML.gif
      (3.56)

      where http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq575_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq576_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq577_HTML.gif are introduced in Lemma 3.5.

      Proposition 3.8.

      Assume that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq578_HTML.gif satisfies the conditions http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq579_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq580_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq581_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq582_HTML.gif . Then there exists a small number http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq583_HTML.gif such that for any http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq584_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq585_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq586_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq587_HTML.gif , there exist at least two nontrivial critical points http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq588_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq589_HTML.gif , in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq590_HTML.gif for the functional http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq591_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ85_HTML.gif
      (3.57)

      where http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq592_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq593_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq594_HTML.gif are introduced in Lemma 3.5.

      Proof.

      It suffices to find the critical points for http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq595_HTML.gif . By Lemma 3.5, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq596_HTML.gif satisfies the torus-sphere variational linking inequality, that is, there exist http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq597_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq598_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq599_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq600_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq601_HTML.gif , and for any http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq602_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq603_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq604_HTML.gif
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ86_HTML.gif
      (3.58)
      By Lemma 3.7, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq605_HTML.gif satisfies the http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq606_HTML.gif condition with respect to http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq607_HTML.gif for every real number http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq608_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ87_HTML.gif
      (3.59)
      Then by Theorem 2.5, there exist two critical points http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq609_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq610_HTML.gif for the functional http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq611_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ88_HTML.gif
      (3.60)
      Setting http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq612_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq613_HTML.gif , we have
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ89_HTML.gif
      (3.61)
      We claim that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq614_HTML.gif , that is http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq615_HTML.gif , which implies that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq616_HTML.gif are the critical points for http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq617_HTML.gif , so http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq618_HTML.gif are the critical points for http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq619_HTML.gif . For this we assume by contradiction that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq620_HTML.gif . From (3.54), http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq621_HTML.gif , namely, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq622_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq623_HTML.gif , are the critical points for http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq624_HTML.gif . By Lemma 3.6, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq625_HTML.gif , which is a contradiction for the fact that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ90_HTML.gif
      (3.62)
      It follows from Lemma 3.6 that there is no critical point http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq626_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ91_HTML.gif
      (3.63)

      Hence, http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq627_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq628_HTML.gif . This proves Proposition 3.8.

      Proof of Theorem 1.1.

      Assume that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq629_HTML.gif satisfies conditions http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq630_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq631_HTML.gif . By Proposition 3.4, there exist http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq632_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq633_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq634_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq635_HTML.gif such that for any http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq636_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq637_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq638_HTML.gif ,(1.1) has at least two nontrivial solutions http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq639_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq640_HTML.gif , in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq641_HTML.gif for the functional http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq642_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ92_HTML.gif
      (3.64)
      By Proposition 3.8, there exist http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq643_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq644_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq645_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq646_HTML.gif such that for any http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq647_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq648_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq649_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq650_HTML.gif , (1.1) has at least two nontrivial solutions http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq651_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq652_HTML.gif , in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq653_HTML.gif for the functional http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq654_HTML.gif such that
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ93_HTML.gif
      (3.65)
      Let
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ94_HTML.gif
      (3.66)

      Then for any http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq655_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq656_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq657_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq658_HTML.gif , (1.1) has at least four nontrivial solutions, two of which are in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq659_HTML.gif and two of which are in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq660_HTML.gif .

      4. Proof of Theorem 1.2

      Assume that http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq661_HTML.gif satisfies conditions http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq662_HTML.gif http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq663_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq664_HTML.gif . Let us set
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ95_HTML.gif
      (4.1)

      Then the space http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq665_HTML.gif is the topological direct sum of the subspaces http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq666_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq667_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq668_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq669_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq670_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq671_HTML.gif are finite dimensional subspaces.

      Proof of Theorem 1.2.

      By the same arguments as that of the proof of Theorem 1.1, there exist http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq672_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq673_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq674_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq675_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq676_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq677_HTML.gif such that for any http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq678_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq679_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq680_HTML.gif , (1.1) has at least four nontrivial solutions, two of which are nontrivial solutions http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq681_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq682_HTML.gif , in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq683_HTML.gif with
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ96_HTML.gif
      (4.2)
      and two of which are nontrivial solutions http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq684_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq685_HTML.gif ,in http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq686_HTML.gif with
      http://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ97_HTML.gif
      (4.3)

      Declarations

      Acknowledgment

      This research is supported in part by Inha University research grant.

      Authors’ Affiliations

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

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      Copyright

      © The Author(s) 2008

      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.