Open Access

Existence of Four Solutions of Some Nonlinear Hamiltonian System

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 https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq2_HTML.gif be a https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq3_HTML.gif function defined on https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq4_HTML.gif which is https://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 https://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 https://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
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ1_HTML.gif
(1.1)
where https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq8_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq9_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ2_HTML.gif
(1.2)
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq10_HTML.gif is the identity matrix on https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq11_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq12_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq13_HTML.gif is the gradient of https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq14_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq15_HTML.gif , https://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
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ3_HTML.gif
(1.3)

We assume that https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq18_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ4_HTML.gif
(1.4)
(H2)Let https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq19_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq20_HTML.gif be integers and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq21_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq23_HTML.gif ) such that https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq24_HTML.gif . Suppose that there exist https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq25_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq26_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq27_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ5_HTML.gif
(1.5)
(H 3) https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq28_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq29_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq30_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ6_HTML.gif
(1.6)

(H4) https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq31_HTML.gif is https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq32_HTML.gif -periodic with respect to https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq34_HTML.gif . The https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq35_HTML.gif -periodic weak solution https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq36_HTML.gif of (1.3) satisfies
https://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
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ8_HTML.gif
(1.8)

where https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq38_HTML.gif satisfies conditions https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq39_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq40_HTML.gif . Then there exists a number https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq41_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq42_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq43_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq44_HTML.gif , https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq47_HTML.gif satisfies conditions https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq48_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq49_HTML.gif . Then there exists a number https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq50_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq51_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq52_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq53_HTML.gif , https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq56_HTML.gif https://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 https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq61_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq62_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq63_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq65_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq66_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq67_HTML.gif on the submanifold with boundary https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq68_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq70_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq72_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq73_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq74_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq75_HTML.gif are the restricted functionals of https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq76_HTML.gif to the manifold with boundary https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq77_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq78_HTML.gif , respectively. Since https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq79_HTML.gif and https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq83_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq85_HTML.gif -periodic solutions in each subspace https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq86_HTML.gif and https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq90_HTML.gif The scalar product in https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq92_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq93_HTML.gif . It is known that if https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq94_HTML.gif is https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq96_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq97_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq98_HTML.gif : https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ10_HTML.gif
(2.2)
,
so that
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ11_HTML.gif
(2.3)
Let https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq100_HTML.gif denote the usual bases in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq101_HTML.gif and set
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ12_HTML.gif
(2.4)
Then https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq102_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq103_HTML.gif are the subspaces of https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq104_HTML.gif on which https://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
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ13_HTML.gif
(2.5)
associated with https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq106_HTML.gif . Here, https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq107_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq108_HTML.gif If https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq109_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq111_HTML.gif . We also note that https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq112_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq113_HTML.gif are mutually orthogonal in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq114_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq115_HTML.gif be the projection from https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq116_HTML.gif onto https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq117_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq118_HTML.gif the one from https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq119_HTML.gif onto https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq120_HTML.gif . Then the norm in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq121_HTML.gif is given by
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq123_HTML.gif , E is compactly embedded in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq124_HTML.gif . In particular, there is an https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq125_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ15_HTML.gif
(2.7)

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

Proposition 2.2.

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

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

Proof.

For https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq136_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ17_HTML.gif
(2.9)
We have
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ18_HTML.gif
(2.10)
Thus, we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq137_HTML.gif is continuous. For https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq138_HTML.gif ,
https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq139_HTML.gif is https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq141_HTML.gif be a Hilbert space and https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq143_HTML.gif such that https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq145_HTML.gif manifold with boundary. Let https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq146_HTML.gif be a https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq147_HTML.gif functional, where https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq148_HTML.gif is an open set containing https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq149_HTML.gif . The https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq152_HTML.gif with the following assumptions: https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq153_HTML.gif where https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq154_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq155_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq156_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq157_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq158_HTML.gif are subspaces of https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq159_HTML.gif ), https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq160_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq161_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq162_HTML.gif are dense in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq163_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq164_HTML.gif , for any https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq167_HTML.gif manifold with boundary in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq168_HTML.gif . We assume that for any https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq169_HTML.gif there exists a retraction https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq170_HTML.gif . For a given https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq171_HTML.gif , we will write https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq172_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq173_HTML.gif be a closed subspace of https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq174_HTML.gif .

Definition 2.3.

Let https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq175_HTML.gif be a closed subset of https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq176_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq177_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq178_HTML.gif be the relative category of https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq179_HTML.gif in https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq181_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq182_HTML.gif , with respect to https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq183_HTML.gif , by
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ21_HTML.gif
(2.13)
We set
https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq184_HTML.gif and assume that

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

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

(3)the https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq187_HTML.gif condition with respect to https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq189_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq190_HTML.gif . If
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ23_HTML.gif
(2.15)
then
https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq192_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq193_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq194_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq195_HTML.gif are three closed subspaces of https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq196_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq197_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq199_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq200_HTML.gif , let https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq201_HTML.gif be the orthogonal projection from https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq202_HTML.gif onto https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq203_HTML.gif . Set
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ25_HTML.gif
(2.17)
and let https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq204_HTML.gif be a https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq205_HTML.gif function defined on a neighborhood https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq206_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq207_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq208_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq209_HTML.gif . One defines
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ26_HTML.gif
(2.18)
Assume that
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ27_HTML.gif
(2.19)
and that the https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq210_HTML.gif condition holds for https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq211_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq212_HTML.gif , with respect to the sequrnce https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq213_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq214_HTML.gif , where
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ28_HTML.gif
(2.20)

Moreover, one assumes https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq215_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq216_HTML.gif has no critical points https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq217_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq218_HTML.gif with https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq220_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq221_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq222_HTML.gif on https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq223_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq224_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq226_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq227_HTML.gif denote the usual bases in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq228_HTML.gif and set
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ29_HTML.gif
(3.1)
Then https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq230_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq231_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq232_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq233_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq234_HTML.gif and https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq236_HTML.gif satisfies the conditions https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq237_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq238_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq239_HTML.gif , and the condition

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

Proof.

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

Since https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq281_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq282_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq283_HTML.gif , there exist a small number https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq284_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq285_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq286_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq287_HTML.gif such that if https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq288_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq289_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq290_HTML.gif . Thus, we have https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq291_HTML.gif . Moreover, if https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq292_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq293_HTML.gif , then we have https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq294_HTML.gif . Thus, https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq297_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq298_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq299_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq300_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq301_HTML.gif is a critical point for https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq302_HTML.gif , then https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq304_HTML.gif , the functional https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq305_HTML.gif is weakly convex in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq306_HTML.gif , while, for fixed https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq307_HTML.gif , the functional https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq308_HTML.gif is strictly concave in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq309_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq310_HTML.gif is the critical point in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq311_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq312_HTML.gif . So if https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq313_HTML.gif is another critical point for https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq314_HTML.gif , then we have
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ35_HTML.gif
(3.7)

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

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

We note that https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq329_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq330_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq331_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq333_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq334_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq335_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq336_HTML.gif , respectively.

Lemma 3.3.

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

Proof.

Let https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq341_HTML.gif be a sequence such that https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq342_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq343_HTML.gif be a sequence in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq344_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq345_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq346_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq347_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq348_HTML.gif . Set https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq349_HTML.gif (and hence https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq350_HTML.gif ) and https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq352_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq353_HTML.gif . Since for https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq354_HTML.gif large https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq355_HTML.gif , we have
https://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),
https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq356_HTML.gif . In the second case, https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq357_HTML.gif . We claim that https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq359_HTML.gif and set https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq360_HTML.gif . Up to a subsequence https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq361_HTML.gif weakly for some https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq362_HTML.gif . By the asymptotically linearity of https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq363_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ45_HTML.gif
(3.17)
We have
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ46_HTML.gif
(3.18)
where https://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
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ47_HTML.gif
(3.19)
Since https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq365_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq366_HTML.gif are bounded and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq367_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq368_HTML.gif , https://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
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ48_HTML.gif
(3.20)
Moreover, we have
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ49_HTML.gif
(3.21)
Since https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq370_HTML.gif converges to 0 weakly and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq371_HTML.gif is bounded, https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq372_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq373_HTML.gif , https://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, https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq376_HTML.gif converges to https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq377_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq378_HTML.gif . We claim that https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq379_HTML.gif converges to https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq380_HTML.gif strongly. We have
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ50_HTML.gif
(3.22)
By https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq381_HTML.gif and the boundedness of https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq382_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ51_HTML.gif
(3.23)
That is, https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq383_HTML.gif converges. Since https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq384_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq385_HTML.gif converges, so https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq386_HTML.gif converges to https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq387_HTML.gif strongly. Therefore, we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq388_HTML.gif , that is, https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq389_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq390_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq391_HTML.gif . In this case, https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq392_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq394_HTML.gif satisfies the conditions https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq395_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq396_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq397_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq398_HTML.gif . Then there exists a number https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq399_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq400_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq401_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq402_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq404_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq405_HTML.gif , in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq406_HTML.gif for the functional https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq407_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ53_HTML.gif
(3.25)

where https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq408_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq409_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq411_HTML.gif . By Lemma 3.1, https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq413_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq414_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq415_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq416_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq417_HTML.gif , and for any https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq418_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq419_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq420_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq421_HTML.gif
https://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, https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq422_HTML.gif satisfies the https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq423_HTML.gif condition with respect to https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq424_HTML.gif for every real number https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq425_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq426_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq427_HTML.gif for the functional https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq428_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ56_HTML.gif
(3.28)
Setting https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq429_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq430_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ57_HTML.gif
(3.29)
We claim that https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq431_HTML.gif , that is https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq432_HTML.gif , which implies that https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq433_HTML.gif are the critical points for https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq434_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq435_HTML.gif , so https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq436_HTML.gif are the critical points for https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq437_HTML.gif in https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq439_HTML.gif . From (3.12), https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq440_HTML.gif , namely, https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq441_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq442_HTML.gif , are the critical points for https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq443_HTML.gif . By Lemma 3.2, https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq445_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ59_HTML.gif
(3.31)

Hence, https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq446_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq448_HTML.gif satisfies the conditions https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq449_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq450_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq451_HTML.gif , and the condition

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

Proof.

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

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

Moreover, if https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq495_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq496_HTML.gif . Thus we have https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq498_HTML.gif there exists a constant https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq499_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq500_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq501_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq502_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq503_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq504_HTML.gif is a critical point for https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq505_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq506_HTML.gif , then https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq508_HTML.gif , a sequence https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq509_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq510_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq511_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq512_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq513_HTML.gif , and a sequence https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq514_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq515_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq516_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq517_HTML.gif . We claim that https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq519_HTML.gif , let us set https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq521_HTML.gif weakly for some https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq522_HTML.gif . Furthermore,
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ64_HTML.gif
(3.36)
so we have
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ65_HTML.gif
(3.37)
Moreover,
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ66_HTML.gif
(3.38)
so we have
https://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
https://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
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ69_HTML.gif
(3.41)
We also have
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ70_HTML.gif
(3.42)
Dividing by https://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
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ71_HTML.gif
(3.43)
Thus
https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq524_HTML.gif . So https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq526_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq527_HTML.gif . From (3.42), we have
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ73_HTML.gif
(3.45)
From (3.40),
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ74_HTML.gif
(3.46)
Thus, https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq528_HTML.gif converges to https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq529_HTML.gif strongly. We claim that https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq530_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq531_HTML.gif . By (H 1) https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq532_HTML.gif , for some https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq533_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq534_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq535_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq536_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq537_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq538_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ75_HTML.gif
(3.47)
If https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq539_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq540_HTML.gif , and
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ76_HTML.gif
(3.48)
Thus, we have
https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq541_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq542_HTML.gif . Thus https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq543_HTML.gif . We proved the lemma.

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

We note that https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq557_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq558_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq559_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq561_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq562_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq563_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq565_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq566_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq567_HTML.gif instead of the space https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq568_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq569_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq570_HTML.gif .

Lemma 3.7.

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

where https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq575_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq576_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq578_HTML.gif satisfies the conditions https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq579_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq580_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq581_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq583_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq584_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq585_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq586_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq588_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq589_HTML.gif , in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq590_HTML.gif for the functional https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq591_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ85_HTML.gif
(3.57)

where https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq592_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq593_HTML.gif , and https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq595_HTML.gif . By Lemma 3.5, https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq597_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq598_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq599_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq600_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq601_HTML.gif , and for any https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq602_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq603_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq604_HTML.gif
https://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, https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq605_HTML.gif satisfies the https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq606_HTML.gif condition with respect to https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq607_HTML.gif for every real number https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq608_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq609_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq610_HTML.gif for the functional https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq611_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ88_HTML.gif
(3.60)
Setting https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq612_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq613_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ89_HTML.gif
(3.61)
We claim that https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq614_HTML.gif , that is https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq615_HTML.gif , which implies that https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq616_HTML.gif are the critical points for https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq617_HTML.gif , so https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq618_HTML.gif are the critical points for https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq620_HTML.gif . From (3.54), https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq621_HTML.gif , namely, https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq622_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq623_HTML.gif , are the critical points for https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq624_HTML.gif . By Lemma 3.6, https://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
https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq626_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ91_HTML.gif
(3.63)

Hence, https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq627_HTML.gif , https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq629_HTML.gif satisfies conditions https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq630_HTML.gif https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq632_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq633_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq634_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq635_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq636_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq637_HTML.gif with https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq639_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq640_HTML.gif , in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq641_HTML.gif for the functional https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq642_HTML.gif such that
https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq643_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq644_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq645_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq646_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq647_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq648_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq649_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq651_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq652_HTML.gif , in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq653_HTML.gif for the functional https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq654_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ93_HTML.gif
(3.65)
Let
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ94_HTML.gif
(3.66)

Then for any https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq655_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq656_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq657_HTML.gif and https://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 https://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 https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq661_HTML.gif satisfies conditions https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq662_HTML.gif https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq663_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq664_HTML.gif . Let us set
https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_Equ95_HTML.gif
(4.1)

Then the space https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq666_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq667_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq668_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq669_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq670_HTML.gif and https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq672_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq673_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq674_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq675_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq676_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq677_HTML.gif such that for any https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq678_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq679_HTML.gif with https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq681_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq682_HTML.gif , in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq683_HTML.gif with
https://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 https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq684_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq685_HTML.gif ,in https://static-content.springer.com/image/art%3A10.1155%2FBVP%2F2008%2F293987/MediaObjects/13661_2007_Article_798_IEq686_HTML.gif with
https://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

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