Open Access

Existence of Positive, Negative, and Sign-Changing Solutions to Discrete Boundary Value Problems

Boundary Value Problems20112011:172818

DOI: 10.1155/2011/172818

Received: 11 November 2010

Accepted: 15 February 2011

Published: 10 March 2011

Abstract

By using critical point theory, Lyapunov-Schmidt reduction method, and characterization of the Brouwer degree of critical points, sufficient conditions to guarantee the existence of five or six solutions together with their sign properties to discrete second-order two-point boundary value problem are obtained. An example is also given to demonstrate our main result.

1. Introduction

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq2_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq3_HTML.gif denote the sets of all natural numbers, integers, and real numbers, respectively. For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq4_HTML.gif , define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq5_HTML.gif , when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq6_HTML.gif . https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq7_HTML.gif is the forward difference operator defined by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq8_HTML.gif .

Consider the following discrete second-order two-point boundary value problem (BVP for short):
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ1_HTML.gif
(1.1)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq9_HTML.gif is a given integer.

By a solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq10_HTML.gif to the BVP (1.1), we mean a real sequence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq11_HTML.gif satisfying (1.1). For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq12_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq13_HTML.gif , we say that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq14_HTML.gif if there exists at least one https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq15_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq16_HTML.gif . We say that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq17_HTML.gif is positive (and write https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq18_HTML.gif ) if for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq19_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq20_HTML.gif  :  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq21_HTML.gif , and similarly, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq22_HTML.gif is negative ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq23_HTML.gif ) if for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq24_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq26_HTML.gif . We say that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq27_HTML.gif is sign-changing if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq28_HTML.gif is neither positive nor negative. Under convenient assumptions, we will prove the existence of five or six solutions to (1.1), which include positive, negative, and sign-changing solutions.

Difference BVP has widely occurred as the mathematical models describing real-life situations in mathematical physics, finite elasticity, combinatorial analysis, and so forth; for example, see [1, 2]. And many scholars have investigated difference BVP independently mainly for two reasons. The first one is that the behavior of discrete systems is sometimes sharply different from the behavior of the corresponding continuous systems. For example, every solution of logistic equation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq29_HTML.gif is monotone, but its discrete analogue https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq30_HTML.gif has chaotic solutions; see [3] for details. The second one is that there is a fundamental relationship between solutions to continuous systems and the corresponding discrete systems by employing discrete variable methods [1]. The classical results on difference BVP employs numerical analysis and features from the linear and nonlinear operator theory, such as fixed point theorems. We remark that, usually, the application of the fixed point theorems yields existence results only.

Recently, however, a few scholars have used critical point theory to deal with the existence of multiple solutions to difference BVP. For example, in 2004, Agarwal et al. [4] employed the mountain pass lemma to study (1.1) with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq31_HTML.gif and obtained the existence of multiple solutions. Very recently, Zheng and Zhang [5] obtained the existence of exactly three solutions to (1.1) by making use of three-critical-point theorem and analytic techniques. We also refer to [69] for more results on the difference BVP by using critical point theory. The application of critical point theory to difference BVP represents an important advance as it allows to prove multiplicity results as well.

Here, by using critical point theory again, as well as Lyapunov-Schmidt reduction method and degree theory, a sharp condition to guarantee the existence of five or six solutions together with their sign properties to (1.1) is obtained. And this paper offers, to the best of our knowledge, a new method to deal with the sign of solutions in the discrete case.

Here, we assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq32_HTML.gif and
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ2_HTML.gif
(1.2)

Hence, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq33_HTML.gif grows asymptotically linear at infinity.

The solvability of (1.1) depends on the properties of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq34_HTML.gif both at zero and at infinity. If
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ3_HTML.gif
(1.3)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq35_HTML.gif is one of the eigenvalues of the eigenvalue problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ4_HTML.gif
(1.4)

then we say that (1.1) is resonant at infinity (or at zero); otherwise, we say that (1.1) is nonresonant at infinity (or at zero). On the eigenvalue problem (1.4), the following results hold (see [1] for details).

Proposition 1.1.

For the eigenvalue problem (1.4), the eigenvalues are
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ5_HTML.gif
(1.5)

and the corresponding eigenfunctions with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq36_HTML.gif are https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq37_HTML.gif .

Remark 1.2.
  1. (i)
    The set of functions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq38_HTML.gif is orthogonal on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq39_HTML.gif with respect to the weight function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq40_HTML.gif ; that is,
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ6_HTML.gif
    (1.6)
     
Moreover, for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq41_HTML.gif .
  1. (ii)

    It is easy to see that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq42_HTML.gif is positive and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq43_HTML.gif changes sign for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq44_HTML.gif ; that is, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq45_HTML.gif  :  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq46_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq47_HTML.gif  :  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq48_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq49_HTML.gif .

     

The main result of this paper is as follows.

Theorem 1.3.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq50_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq51_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq52_HTML.gif , then (1.1) has at least five solutions. Moreover, one of the following cases occurs:
  1. (i)

       https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq53_HTML.gif is even and (1.1) has two sign-changing solutions,

     
  2. (ii)

       https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq54_HTML.gif is even and (1.1) has six solutions, three of which are of the same sign,

     
  3. (iii)

       https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq55_HTML.gif is odd and (1.1) has two sigh-changing solutions,

     
  4. (iv)

       https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq56_HTML.gif is odd and (1.1) has three solutions of the same sign.

     

Remark 1.4.

The assumption https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq57_HTML.gif in Theorem 1.3 is sharp in the sense that when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq58_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq59_HTML.gif , Theorem 1.4 of [5] gives sufficient conditions for (1.1) to have exactly three solutions with some restrictive conditions.

Example 1.5.

Consider the BVP
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ7_HTML.gif
(1.7)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq60_HTML.gif is defined as follows:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ8_HTML.gif
(1.8)

It is easy to verify that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq61_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq62_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq63_HTML.gif . So, all the conditions in Theorem 1.3 are satisfied with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq64_HTML.gif . And hence (1.7) has at least five solutions, among which two sign-changing solutions or three solutions of the same sign.

By the computation of critical groups, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq65_HTML.gif , we have the following.

Corollary 1.6 (see Remark 3.7 below).

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq66_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq67_HTML.gif , then (1.1) has at least one positive solution and one negative solution.

2. Preliminaries

Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ9_HTML.gif
(2.1)
Then, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq68_HTML.gif is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq69_HTML.gif -dimensional Hilbert space with inner product
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ10_HTML.gif
(2.2)
by which the norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq70_HTML.gif can be induced by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ11_HTML.gif
(2.3)

Here, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq71_HTML.gif denotes the Euclidean norm in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq72_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq73_HTML.gif denotes the usual inner product in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq74_HTML.gif .

Define
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ12_HTML.gif
(2.4)
Then, the functional https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq75_HTML.gif is of class https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq76_HTML.gif with
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ13_HTML.gif
(2.5)

So, solutions to (1.1) are precisely the critical points of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq77_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq78_HTML.gif .

As we have mentioned, we will use critical point theory, Lyapunov-Schmidt reduction method, and degree theory to prove our result. Let us collect some results that will be used below. One can refer to [1012] for more details.

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq79_HTML.gif be a Hilbert space and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq80_HTML.gif . Denote
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ14_HTML.gif
(2.6)

for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq81_HTML.gif . The following is the definition of the Palais-Smale (PS) compactness condition.

Definition 2.1.

The functional https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq82_HTML.gif satisfies the (PS) condition if any sequence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq83_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq84_HTML.gif is bounded and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq85_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq86_HTML.gif has a convergent subsequence.

In [13], Cerami introduced a weak version of the (PS) condition as follows.

Definition 2.2.

The functional https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq87_HTML.gif satisfies the Cerami (C) condition if any sequence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq88_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq89_HTML.gif is bounded and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq90_HTML.gif , as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq91_HTML.gif has a convergent subsequence.

If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq92_HTML.gif satisfies the (PS) condition or the (C) condition, then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq93_HTML.gif satisfies the following deformation condition which is essential in critical point theory (cf. [14, 15]).

Definition 2.3.

The functional https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq94_HTML.gif satisfies the ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq95_HTML.gif ) condition at the level https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq96_HTML.gif if for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq97_HTML.gif and any neighborhood N of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq99_HTML.gif , there are https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq100_HTML.gif and a continuous deformation https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq101_HTML.gif such that
  1. (i)

       https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq102_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq103_HTML.gif ,

     
  2. (ii)

       https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq104_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq105_HTML.gif ,

     
  3. (iii)

        https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq106_HTML.gif is non-increasing in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq107_HTML.gif for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq108_HTML.gif ,

     
  4. (iv)

       https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq109_HTML.gif .

     

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq110_HTML.gif satisfies the (D) condition if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq111_HTML.gif satisfies the ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq112_HTML.gif ) condition for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq113_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq114_HTML.gif denote singular homology with coefficients in a field https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq115_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq116_HTML.gif is a critical point of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq117_HTML.gif with critical level https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq118_HTML.gif , then the critical groups of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq119_HTML.gif are defined by
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ15_HTML.gif
(2.7)
Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq120_HTML.gif is strictly bounded from below by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq121_HTML.gif and that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq122_HTML.gif satisfies ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq123_HTML.gif ) for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq124_HTML.gif . Then, the https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq125_HTML.gif th critical group at infinity of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq126_HTML.gif is defined in [16] as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ16_HTML.gif
(2.8)

Due to the condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq127_HTML.gif , these groups are not dependent on the choice of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq128_HTML.gif .

Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq129_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq130_HTML.gif satisfies the (D) condition. The Morse-type numbers of the pair https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq131_HTML.gif are defined by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq132_HTML.gif , and the Betti numbers of the pair https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq133_HTML.gif are defined by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq134_HTML.gif . By Morse theory [10, 11], the following relations hold:
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ17_HTML.gif
(2.9)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ18_HTML.gif
(2.10)

It follows that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq135_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq136_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq137_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq138_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq139_HTML.gif . Thus, when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq140_HTML.gif for some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq141_HTML.gif must have a critical point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq142_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq143_HTML.gif .

The critical groups of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq144_HTML.gif at an isolated critical point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq145_HTML.gif describe the local behavior of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq146_HTML.gif near https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq147_HTML.gif , while the critical groups of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq148_HTML.gif at infinity describe the global property of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq149_HTML.gif . In most applications, unknown critical points will be found from (2.9) or (2.10) if we can compute both the critical groups at known critical points and the critical groups at infinity. Thus, the computation of the critical groups is very important. Now, we collect some useful results on computation of critical groups which will be employed in our discussion.

Proposition 2.4 (see [16]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq150_HTML.gif be a real Hilbert space and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq151_HTML.gif . Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq152_HTML.gif splits as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq153_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq154_HTML.gif is bounded from below on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq155_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq156_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq157_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq158_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq159_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq160_HTML.gif .

Proposition 2.5 (see [17]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq161_HTML.gif be a separable Hilbert space with inner product https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq162_HTML.gif and corresponding norm https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq163_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq164_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq165_HTML.gif closed subspaces of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq166_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq167_HTML.gif . Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq168_HTML.gif satisfies the (PS) condition and the critical values of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq169_HTML.gif are bounded from below. If there is a real number https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq170_HTML.gif such that for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq171_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq172_HTML.gif , there holds
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ19_HTML.gif
(2.11)
then there exists a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq173_HTML.gif -functional https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq174_HTML.gif  :  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq175_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ20_HTML.gif
(2.12)

Moreover, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq176_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq177_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq178_HTML.gif .

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq179_HTML.gif denote the open ball in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq180_HTML.gif about 0 of the radius https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq181_HTML.gif , and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq182_HTML.gif denote its boundary.

Lemma 2.6 (Mountain Pass Lemma [10, 11]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq183_HTML.gif be a real Banach space and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq184_HTML.gif satisfying the (PS) condition. Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq185_HTML.gif and

(J1)  there are constants https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq186_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq187_HTML.gif , and

(J2)  there is a https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq188_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq189_HTML.gif .

Then, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq190_HTML.gif possesses a critical value https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq191_HTML.gif . Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq192_HTML.gif can be characterized as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ21_HTML.gif
(2.13)
where
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ22_HTML.gif
(2.14)

Definition 2.7 (Mountain pass point).

An isolated critical point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq193_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq194_HTML.gif is called a mountain pass point if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq195_HTML.gif .

To compute the critical groups of a mountain pass point, we have the following result.

Proposition 2.8 (see [11]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq196_HTML.gif be a real Hilbert space. Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq197_HTML.gif has a mountain pass point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq198_HTML.gif and that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq199_HTML.gif is a Fredholm operator with finite Morse index satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ23_HTML.gif
(2.15)
Then,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ24_HTML.gif
(2.16)

The following theorem gives a relation between the Leray-Schauder degree and the critical groups.

Theorem 2.9 (see [10, 11]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq200_HTML.gif be a real Hilbert space, and let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq201_HTML.gif be a function satisfying the (PS) condition. Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq202_HTML.gif , where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq203_HTML.gif  :  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq204_HTML.gif is a completely continuous operator. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq205_HTML.gif is an isolated critical point of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq206_HTML.gif , that is, there exists a neighborhood https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq207_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq208_HTML.gif , such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq209_HTML.gif is the only critical point of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq210_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq211_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ25_HTML.gif
(2.17)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq212_HTML.gif denotes the Leray-Schauder degree.

Finally, we state a global version of the Lyapunov-Schmidt reduction method.

Lemma 2.10 (see [18]).

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq213_HTML.gif be a real separable Hilbert space. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq214_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq215_HTML.gif be closed subspaces of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq216_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq217_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq218_HTML.gif . If there are https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq219_HTML.gif such that for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq220_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ26_HTML.gif
(2.18)

then the following results hold.

(i)  There exists a continuous function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq221_HTML.gif  :  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq222_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ27_HTML.gif
(2.19)

Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq223_HTML.gif is the unique member of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq224_HTML.gif such that

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ28_HTML.gif
(2.20)

(ii)  The function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq225_HTML.gif  :  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq226_HTML.gif defined by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq227_HTML.gif is of class https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq228_HTML.gif , and

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ29_HTML.gif
(2.21)

(iii)  An element https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq229_HTML.gif is a critical point of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq230_HTML.gif if and only if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq231_HTML.gif is a critical point of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq232_HTML.gif .

(iv)  Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq233_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq234_HTML.gif be the projection onto https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq235_HTML.gif across https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq236_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq237_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq238_HTML.gif be open bounded regions such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ30_HTML.gif
(2.22)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq239_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq240_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ31_HTML.gif
(2.23)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq241_HTML.gif denotes the Leray-Schauder degree.

(v)  If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq242_HTML.gif is a critical point of mountain pass type of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq243_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq244_HTML.gif is a critical point of mountain pass type of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq245_HTML.gif .

3. Proof of Theorem 1.3

In this section, firstly, we obtain a positive solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq246_HTML.gif and a negative solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq247_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq248_HTML.gif to (1.1) by using cutoff technique and the mountain pass lemma. Then, we give a precise computation of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq249_HTML.gif . And we remark that under the assumptions of Theorem 1.3, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq250_HTML.gif can be completely computed by using Propositions 2.4 and 2.5. Based on these results, four nontrivial solutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq251_HTML.gif to (1.1) can be obtained by (2.9) or (2.10). However, it seems difficult to obtain the sign property of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq252_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq253_HTML.gif through their depiction of critical groups. To conquer this difficulty, we compute the Brouwer degree of the sets of positive solutions and negative solutions to (1.1). Finally, the third nontrivial solution to (1.1) is obtained by Lyapunov-Schmidt reduction method, and its characterization of the local degree results in one or two more nontrivial solutions to (1.1) together with their sign property.

Let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ32_HTML.gif
(3.1)
and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq254_HTML.gif . The functionals https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq255_HTML.gif are defined as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ33_HTML.gif
(3.2)

Remark 3.1.

From the definitions of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq256_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq257_HTML.gif , it is easy to see that if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq258_HTML.gif is a critical point of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq259_HTML.gif (or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq260_HTML.gif ), then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq261_HTML.gif (or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq262_HTML.gif ).

Lemma 3.2.

The functionals https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq263_HTML.gif satisfy the (PS) condition; that is, every sequence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq264_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq265_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq266_HTML.gif is bounded, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq267_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq268_HTML.gif has a convergent subsequence.

Proof.

We only prove the case of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq269_HTML.gif . The case of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq270_HTML.gif is completely similar. Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq271_HTML.gif is finite dimensional, it suffices to show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq272_HTML.gif is bounded. Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq273_HTML.gif is unbounded. Passing to a subsequence, we may assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq274_HTML.gif and for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq275_HTML.gif , either https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq276_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq277_HTML.gif is bounded.

Set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq278_HTML.gif . For a subsequence, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq279_HTML.gif converges to some https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq280_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq281_HTML.gif . Since for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq282_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ34_HTML.gif
(3.3)
Hence,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ35_HTML.gif
(3.4)
If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq283_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ36_HTML.gif
(3.5)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq284_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq285_HTML.gif . If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq286_HTML.gif is bounded, then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ37_HTML.gif
(3.6)
Letting https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq287_HTML.gif in (3.4), we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ38_HTML.gif
(3.7)
which implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq288_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ39_HTML.gif
(3.8)

Because https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq289_HTML.gif , we see that if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq290_HTML.gif is a solution to (3.8), then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq291_HTML.gif is positive. Since this contradicts https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq292_HTML.gif , we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq293_HTML.gif is the only solution to (3.8). A contradiction to https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq294_HTML.gif .

Lemma 3.3.

Under the conditions of Theorem 1.3, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq295_HTML.gif has a positive mountain pass-type critical point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq296_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq298_HTML.gif has a negative mountain pass-type critical point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq299_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq300_HTML.gif .

Proof.

We only prove the case of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq301_HTML.gif . Firstly, we will prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq302_HTML.gif satisfies all the conditions in Lemma 2.6. And hence, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq303_HTML.gif has at least one nonzero critical point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq304_HTML.gif . In fact, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq305_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq306_HTML.gif satisfies the (PS) condition by Lemma 3.2. Clearly, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq307_HTML.gif . Thus, we still have to show that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq308_HTML.gif satisfies (J1), (J2). To verify (J1), set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq309_HTML.gif , then for any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq310_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq311_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ40_HTML.gif
(3.9)
So, by Taylor series expansion,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ41_HTML.gif
(3.10)
Take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq312_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq313_HTML.gif . If we set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq314_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ42_HTML.gif
(3.11)
Since for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq315_HTML.gif , if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq316_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq317_HTML.gif for every https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq318_HTML.gif and hence
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ43_HTML.gif
(3.12)
where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq319_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq320_HTML.gif . If we take
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ44_HTML.gif
(3.13)

then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq321_HTML.gif . And hence, (J1) holds.

To verify (J2), note that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq322_HTML.gif implies that there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq323_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq324_HTML.gif , such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ45_HTML.gif
(3.14)
So, if we take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq325_HTML.gif with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq326_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ46_HTML.gif
(3.15)

So, if we take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq327_HTML.gif sufficiently large such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq328_HTML.gif and for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq329_HTML.gif , then (J2) holds.

Now, by Lemma 2.6, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq330_HTML.gif has at least a nonzero critical point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq331_HTML.gif . And for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq332_HTML.gif , we claim that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq333_HTML.gif . If not, set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq334_HTML.gif , then for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq335_HTML.gif . By https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq336_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq337_HTML.gif . Hence, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq338_HTML.gif .

In the following, we will compute the critical groups https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq339_HTML.gif by using Proposition 2.8.

Assume that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ47_HTML.gif
(3.16)
and that there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq340_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ48_HTML.gif
(3.17)
This implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq341_HTML.gif satisfies
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ49_HTML.gif
(3.18)
Hence, the eigenvalue problem
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ50_HTML.gif
(3.19)
has an eigenvalue https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq342_HTML.gif . Condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq343_HTML.gif implies that 1 must be a simple eigenvalue; see [1]. So, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq344_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq345_HTML.gif is finite dimensional, the Morse index of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq346_HTML.gif must be finite and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq347_HTML.gif must be a Fredholm operator. By Proposition 2.8, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq348_HTML.gif . Finally, choose the neighborhood https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq349_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq350_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq351_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq352_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ51_HTML.gif
(3.20)

The proof is complete.

Lemma 3.4.

By https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq353_HTML.gif , one has
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ52_HTML.gif
(3.21)

Proof.

By assumption, we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq354_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq355_HTML.gif ,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ53_HTML.gif
(3.22)

which implies that 0 is a local minimizer of both https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq356_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq357_HTML.gif . Hence, (3.21) holds.

Remark 3.5.

Under the conditions of Theorem 1.3, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ54_HTML.gif
(3.23)
We will use Propositions 2.4 and 2.5 to prove (3.23). Very similar to the proof of Lemma 3.2, we can prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq358_HTML.gif satisfies the (PS) condition. And it is easy to prove that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq359_HTML.gif satisfies (2.11). In fact, let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ55_HTML.gif
(3.24)
By https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq360_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq361_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq362_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ56_HTML.gif
(3.25)

Hence, if we set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq363_HTML.gif , then (2.11) holds.

Now, noticing that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq364_HTML.gif implies that there exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq365_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq366_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq367_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ57_HTML.gif
(3.26)
Hence, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ58_HTML.gif
(3.27)
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ59_HTML.gif
(3.28)

Then, (3.23) is proved by Propositions 2.4 and 2.5.

Remark 3.6.

Following the proof of Theorem 3.1 in [17], (3.23) implies that there must exist a critical point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq368_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq369_HTML.gif satisfying
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ60_HTML.gif
(3.29)
It is known that the critical groups are useful in distinguishing critical points. So far, we have obtained four critical points 0, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq370_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq371_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq372_HTML.gif together with their characterization of critical groups. Assume that 0, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq373_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq374_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq375_HTML.gif are the only critical points of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq376_HTML.gif . Then, the Morse inequality (2.10) becomes
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ61_HTML.gif
(3.30)

This is impossible. Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq377_HTML.gif must have at least one more critical point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq378_HTML.gif . Hence, (1.1) has at least five solutions. However, it seems difficult to obtain the sign property of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq379_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq380_HTML.gif . To obtain more refined results, we seek the third nontrivial solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq381_HTML.gif to (1.1) by Lyapunov-Schmidt reduction method and then its characterization of the local degree results in one or two more nontrivial solutions to (1.1) together with their sign property.

Remark 3.7.

The condition https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq382_HTML.gif in Theorem 1.3 is necessary to obtain three or more nontrivial solutions to (1.1). In fact, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq383_HTML.gif , then we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ62_HTML.gif
(3.31)
Hence, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq384_HTML.gif may coincide with https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq385_HTML.gif or https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq386_HTML.gif which becomes an obstacle to seek other critical points by using Morse inequality. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq387_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ63_HTML.gif
(3.32)

Hence, one cannot exclude the possibility of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq388_HTML.gif .

To compute the degree of the set of positive (or negative) solutions to (1.1), we need the following lemma.

Lemma 3.8.

There exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq389_HTML.gif large enough, such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ64_HTML.gif
(3.33)

Proof.

We only prove the case of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq390_HTML.gif . For any https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq391_HTML.gif , define https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq392_HTML.gif as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ65_HTML.gif
(3.34)
Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq393_HTML.gif . The functional https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq394_HTML.gif  :  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq395_HTML.gif is defined as
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ66_HTML.gif
(3.35)
It is obvious that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq396_HTML.gif is of class https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq397_HTML.gif and its critical points are precisely solutions to
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ67_HTML.gif
(3.36)

Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq398_HTML.gif , we see that if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq399_HTML.gif is a solution to (3.36), then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq400_HTML.gif is positive. Because this contradicts https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq401_HTML.gif , we conclude that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq402_HTML.gif is the only critical point of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq403_HTML.gif .

We claim that if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq404_HTML.gif is a ball in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq405_HTML.gif containing zero, then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq406_HTML.gif . In fact, since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq407_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq408_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq409_HTML.gif . Hence, for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq410_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ68_HTML.gif
(3.37)
where we have used the fact that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq411_HTML.gif is positive on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq412_HTML.gif . Then, for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq413_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq414_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ69_HTML.gif
(3.38)
Hence, by invariance under homotopy of Brouwer degree, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ70_HTML.gif
(3.39)

where https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq415_HTML.gif .

Now, let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq416_HTML.gif . We claim that for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq417_HTML.gif large enough and for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq418_HTML.gif , the function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq419_HTML.gif has no zero on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq420_HTML.gif .

In fact, we have proved that for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq421_HTML.gif and for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq422_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ71_HTML.gif
(3.40)
On the other hand, by the definition of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq423_HTML.gif , for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq424_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq425_HTML.gif large enough such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq426_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq427_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq428_HTML.gif , for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq429_HTML.gif , take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq430_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ72_HTML.gif
(3.41)
For https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq431_HTML.gif , take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq432_HTML.gif , then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ73_HTML.gif
(3.42)
Hence, if we take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq433_HTML.gif , then for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq434_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq435_HTML.gif , and for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq436_HTML.gif , we have https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq437_HTML.gif . So, if we let
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ74_HTML.gif
(3.43)
then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq438_HTML.gif for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq439_HTML.gif . And for all https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq440_HTML.gif , we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ75_HTML.gif
(3.44)
So far, we have proved that for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq441_HTML.gif large enough, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq442_HTML.gif has no zero point on https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq443_HTML.gif for each https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq444_HTML.gif . Hence, by invariance under homotopy of Brouwer degree, we obtain
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ76_HTML.gif
(3.45)

This completes the proof.

Remark 3.9.

By Theorem 2.9 and the above results, we have the following characterization of degree of critical points.

  If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq446_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq447_HTML.gif ) is a neighborhood of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq448_HTML.gif ( https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq449_HTML.gif ) containing no other critical points, then

https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ77_HTML.gif
(3.46)
  Assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq451_HTML.gif is a ball centered at zero containing on other critical points, then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ78_HTML.gif
(3.47)
Hence, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq452_HTML.gif is a bounded region containing the positive critical points and no other critical points, then by (3.33) we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ79_HTML.gif
(3.48)
Similarly, we see that if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq453_HTML.gif is a bounded region containing the negative critical points and no other critical points, then
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ80_HTML.gif
(3.49)

Now, we can give the proof of Theorem 1.3.

Proof of Theorem 1.3.

The functional https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq454_HTML.gif satisfies (2.18) in Lemma 2.10 due to the fact that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq455_HTML.gif satisfies (2.11). Hence, by Lemma 2.10, there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq456_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ81_HTML.gif
(3.50)
Moreover, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq458_HTML.gif is the unique member of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq459_HTML.gif such that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ82_HTML.gif
(3.51)
The function https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq460_HTML.gif defined by https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq462_HTML.gif is of class https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq463_HTML.gif . Because https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq464_HTML.gif , (3.27) implies that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq465_HTML.gif as https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq466_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq467_HTML.gif , there must exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq468_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq469_HTML.gif . Take https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq470_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq471_HTML.gif by (iii) of Lemma 2.10. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq472_HTML.gif is a neighborhood of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq473_HTML.gif containing no other critical points of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq474_HTML.gif , taking https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq475_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq476_HTML.gif . Then, by part (iv) of Lemma 2.10, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ83_HTML.gif
(3.52)

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq477_HTML.gif Is Even

Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq478_HTML.gif be large enough so that if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq479_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq480_HTML.gif . Because https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq481_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq482_HTML.gif is of class https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq483_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq484_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq485_HTML.gif for https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq486_HTML.gif . Because https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq487_HTML.gif is coercive, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq488_HTML.gif . Hence, if we set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq489_HTML.gif  :  https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq490_HTML.gif , then by (iv) of Lemma 2.10, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ84_HTML.gif
(3.53)
Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq491_HTML.gif is finite. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq492_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq493_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq494_HTML.gif be disjoint open bounded regions in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq495_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq496_HTML.gif is the set of positive critical points of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq497_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq498_HTML.gif is the set of negative critical points of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq499_HTML.gif . So far, we have proved that
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ85_HTML.gif
(3.54)
(i) If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq500_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq501_HTML.gif is sign changing. Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq502_HTML.gif denote an open bounded region disjoint from https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq503_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq504_HTML.gif . By the excision property of Brouwer degree, we have
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ86_HTML.gif
(3.55)
Thus, by Kronecker existence property of Brouwer degree, we see that there must exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq505_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq506_HTML.gif , which proves that (1.1) has at least five solutions. In this case, both https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq507_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq508_HTML.gif change sign.
  1. (ii)
    Suppose now that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq509_HTML.gif . Without loss of generality, we may assume that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq510_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq511_HTML.gif be a neighborhood of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq512_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq513_HTML.gif . By Lemma 3.3, there exists a critical point of mountain pass type https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq514_HTML.gif such that if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq515_HTML.gif is a neighborhood of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq516_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq517_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq518_HTML.gif . Thus,
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ87_HTML.gif
    (3.56)
     
Thus, by Kronecker existence property of Brouwer degree, there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq519_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq520_HTML.gif . Finally,
https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ88_HTML.gif
(3.57)

Thus, there must exist https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq521_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq522_HTML.gif . Thus, the set https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq523_HTML.gif together with a critical point https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq524_HTML.gif of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq525_HTML.gif in https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq526_HTML.gif shows that (1.1) has five nontrivial solutions. Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq527_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq528_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq529_HTML.gif is a sign-changing solution, and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq530_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq531_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq532_HTML.gif have the same sign. This completes the proof of Theorem 1.3, when https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq533_HTML.gif is even.

Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq534_HTML.gif Is Odd
  1. (iii)

    Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq535_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq536_HTML.gif , and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq537_HTML.gif be as above. If https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq538_HTML.gif , the proof follows very closely that of the case (i).

     
  2. (iv)
    Suppose that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq539_HTML.gif , hence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq540_HTML.gif . Because https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq541_HTML.gif , there exists https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq542_HTML.gif such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq543_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq544_HTML.gif . So, if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq545_HTML.gif , then https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq546_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq547_HTML.gif is a local maximum of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq548_HTML.gif . Since we are assuming (1.1) to have only finitely many solutions, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq549_HTML.gif is a strictly local maximum of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq550_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq551_HTML.gif be such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq552_HTML.gif if https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq553_HTML.gif . Since https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq554_HTML.gif , https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq555_HTML.gif is path connected. Thus, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq556_HTML.gif is not a critical point of mountain pass type. By Lemma 3.3, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq557_HTML.gif has a critical point of mountain pass type https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq558_HTML.gif . By (v) of Lemma 2.10, https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq559_HTML.gif , and hence https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq560_HTML.gif . Let https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq561_HTML.gif be neighborhoods of https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq562_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq563_HTML.gif , respectively, such that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq564_HTML.gif and https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq565_HTML.gif . Thus,
    https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ89_HTML.gif
    (3.58)
     

Thus, by Kronecker existence property of Brouwer degree, there exists a third positive solution https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq566_HTML.gif . So far, we have proved that (1.1) has at least four nontrivial solutions https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq567_HTML.gif and that https://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq568_HTML.gif have the same sign. This proves Theorem 1.3.

Declarations

Acknowledgments

Project supported by National Natural Science Foundation of China (no. 11026059) and Foundation for Distinguished Young Talents in Higher Education of Guangdong, China (no. LYM09105).

Authors’ Affiliations

(1)
School of Mathematics and Information Sciences, Guangzhou University
(2)
Department of Basic Courses, Guangdong Baiyun Institute

References

  1. Agarwal RP: Difference Equations and Inequalities: Theory, Methods, and Applications, Monographs and Textbooks in Pure and Applied Mathematics. Volume 228. 2nd edition. Marcel Dekker, New York, NY, USA; 2000:xvi+971.Google Scholar
  2. Sharkovsky AN, Maĭstrenko YL, Romanenko EY: Difference Equations and Their Applications, Mathematics and Its Applications. Volume 250. Kluwer Academic, Dordrecht, The Netherlands; 1993:xii+358.View ArticleGoogle Scholar
  3. May RM: Simple mathematical models with very complicated dynamics. Nature 1976, 261: 459-466. 10.1038/261459a0View ArticleGoogle Scholar
  4. Agarwal RP, Perera K, O'Regan D: Multiple positive solutions of singular and nonsingular discrete problems via variational methods. Nonlinear Analysis. Theory, Methods & Applications 2004, 58(1-2):69-73. 10.1016/j.na.2003.11.012View ArticleMathSciNetMATHGoogle Scholar
  5. Zheng B, Zhang Q: Existence and multiplicity of solutions of second-order difference boundary value problems. Acta Applicandae Mathematicae 2010, 110(1):131-152. 10.1007/s10440-008-9389-xView ArticleMathSciNetMATHGoogle Scholar
  6. Cai X, Yu J: Existence theorems for second-order discrete boundary value problems. Journal of Mathematical Analysis and Applications 2006, 320(2):649-661. 10.1016/j.jmaa.2005.07.029View ArticleMathSciNetMATHGoogle Scholar
  7. Jiang L, Zhou Z: Existence of nontrivial solutions for discrete nonlinear two point boundary value problems. Applied Mathematics and Computation 2006, 180(1):318-329. 10.1016/j.amc.2005.12.018View ArticleMathSciNetMATHGoogle Scholar
  8. Liang H, Weng P: Existence and multiple solutions for a second-order difference boundary value problem via critical point theory. Journal of Mathematical Analysis and Applications 2007, 326(1):511-520. 10.1016/j.jmaa.2006.03.017View ArticleMathSciNetMATHGoogle Scholar
  9. Aprahamian M, Souroujon D, Tersian S: Decreasing and fast solutions for a second-order difference equation related to Fisher-Kolmogorov's equation. Journal of Mathematical Analysis and Applications 2010, 363(1):97-110. 10.1016/j.jmaa.2009.08.009View ArticleMathSciNetMATHGoogle Scholar
  10. Mawhin J, Willem M: Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sciences. Volume 74. Springer, New York, NY, USA; 1989:xiv+277.View ArticleGoogle Scholar
  11. Chang K-C: Infinite-Dimensional Morse Theory and Multiple Solution Problems. Birkhäuser, Boston, Mass, USA; 1993:x+312.View ArticleMATHGoogle Scholar
  12. Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama, Japan; 2000:iv+276.MATHGoogle Scholar
  13. Cerami G: An existence criterion for the critical points on unbounded manifolds. Istituto Lombardo. Accademia di Scienze e Lettere. Rendiconti A 1978, 112(2):332-336.MathSciNetMATHGoogle Scholar
  14. Bartolo P, Benci V, Fortunato D: Abstract critical point theorems and applications to some nonlinear problems with "strong" resonance at infinity. Nonlinear Analysis. Theory, Methods & Applications 1983, 7(9):981-1012. 10.1016/0362-546X(83)90115-3View ArticleMathSciNetMATHGoogle Scholar
  15. Chang KC: Solutions of asymptotically linear operator equations via Morse theory. Communications on Pure and Applied Mathematics 1981, 34(5):693-712. 10.1002/cpa.3160340503View ArticleMathSciNetMATHGoogle Scholar
  16. Bartsch T, Li SJ: Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Analysis. Theory, Methods & Applications 1997, 28(3):419-441. 10.1016/0362-546X(95)00167-TView ArticleMathSciNetMATHGoogle Scholar
  17. Liu S, Li S: Critical groups at infinity, saddle point reduction and elliptic resonant problems. Communications in Contemporary Mathematics 2003, 5(5):761-773. 10.1142/S0219199703001129View ArticleMathSciNetMATHGoogle Scholar
  18. Castro A, Cossio J: Multiple solutions for a nonlinear Dirichlet problem. SIAM Journal on Mathematical Analysis 1994, 25(6):1554-1561. 10.1137/S0036141092230106View ArticleMathSciNetMATHGoogle Scholar

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© Bo Zheng et al. 2011

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