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

  • Bo Zheng1Email author,

    Affiliated with

    • Huafeng Xiao1 and

      Affiliated with

      • Haiping Shi2

        Affiliated with

        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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq2_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq4_HTML.gif , define http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq5_HTML.gif , when http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq6_HTML.gif . http://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 http://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):
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ1_HTML.gif
        (1.1)

        where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq11_HTML.gif satisfying (1.1). For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq12_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq13_HTML.gif , we say that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq14_HTML.gif if there exists at least one http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq15_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq16_HTML.gif . We say that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq17_HTML.gif is positive (and write http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq18_HTML.gif ) if for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq19_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq20_HTML.gif  :  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq21_HTML.gif , and similarly, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq22_HTML.gif is negative ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq23_HTML.gif ) if for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq24_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq26_HTML.gif . We say that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq27_HTML.gif is sign-changing if http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq29_HTML.gif is monotone, but its discrete analogue http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq32_HTML.gif and
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ2_HTML.gif
        (1.2)

        Hence, http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ3_HTML.gif
        (1.3)
        where http://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
        http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq36_HTML.gif are http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq38_HTML.gif is orthogonal on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq39_HTML.gif with respect to the weight function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq40_HTML.gif ; that is,
          http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ6_HTML.gif
          (1.6)
           
        Moreover, for each http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq42_HTML.gif is positive and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq43_HTML.gif changes sign for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq44_HTML.gif ; that is, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq45_HTML.gif  :  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq46_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq47_HTML.gif  :  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq48_HTML.gif for http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq50_HTML.gif with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq51_HTML.gif , and http://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)

             http://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)

             http://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)

             http://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)

             http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq58_HTML.gif for http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ7_HTML.gif
        (1.7)
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq60_HTML.gif is defined as follows:
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq61_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq62_HTML.gif , and http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq66_HTML.gif , and http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ9_HTML.gif
        (2.1)
        Then, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq68_HTML.gif is a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq69_HTML.gif -dimensional Hilbert space with inner product
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ10_HTML.gif
        (2.2)
        by which the norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq70_HTML.gif can be induced by
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ11_HTML.gif
        (2.3)

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

        Define
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ12_HTML.gif
        (2.4)
        Then, the functional http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq75_HTML.gif is of class http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq76_HTML.gif with
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq77_HTML.gif in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq79_HTML.gif be a Hilbert space and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq80_HTML.gif . Denote
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ14_HTML.gif
        (2.6)

        for http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq83_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq84_HTML.gif is bounded and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq85_HTML.gif as http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq88_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq89_HTML.gif is bounded and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq90_HTML.gif , as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq91_HTML.gif has a convergent subsequence.

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

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

           
        2. (ii)

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

           
        3. (iii)

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

           
        4. (iv)

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

           

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

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

        Due to the condition http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq128_HTML.gif .

        Assume that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq129_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq131_HTML.gif are defined by http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq133_HTML.gif are defined by http://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:
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ17_HTML.gif
        (2.9)
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ18_HTML.gif
        (2.10)

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

        The critical groups of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq144_HTML.gif at an isolated critical point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq145_HTML.gif describe the local behavior of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq146_HTML.gif near http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq147_HTML.gif , while the critical groups of http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq150_HTML.gif be a real Hilbert space and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq151_HTML.gif . Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq152_HTML.gif splits as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq153_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq154_HTML.gif is bounded from below on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq155_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq156_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq157_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq158_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq159_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq160_HTML.gif .

        Proposition 2.5 (see [17]).

        Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq162_HTML.gif and corresponding norm http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq163_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq164_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq165_HTML.gif closed subspaces of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq166_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq167_HTML.gif . Assume that http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq170_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq171_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq172_HTML.gif , there holds
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ19_HTML.gif
        (2.11)
        then there exists a http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq173_HTML.gif -functional http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq174_HTML.gif  :  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq175_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ20_HTML.gif
        (2.12)

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

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq179_HTML.gif denote the open ball in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq180_HTML.gif about 0 of the radius http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq181_HTML.gif , and let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq183_HTML.gif be a real Banach space and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq184_HTML.gif satisfying the (PS) condition. Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq185_HTML.gif and

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

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

        Then, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq190_HTML.gif possesses a critical value http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq191_HTML.gif . Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq192_HTML.gif can be characterized as
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ21_HTML.gif
        (2.13)
        where
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq193_HTML.gif of http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq197_HTML.gif has a mountain pass point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq198_HTML.gif and that http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ23_HTML.gif
        (2.15)
        Then,
        http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq202_HTML.gif , where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq203_HTML.gif  :  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq204_HTML.gif is a completely continuous operator. If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq205_HTML.gif is an isolated critical point of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq206_HTML.gif , that is, there exists a neighborhood http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq207_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq208_HTML.gif , such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq209_HTML.gif is the only critical point of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq210_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq211_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ25_HTML.gif
        (2.17)

        where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq214_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq215_HTML.gif be closed subspaces of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq216_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq217_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq218_HTML.gif . If there are http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq219_HTML.gif such that for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq220_HTML.gif ,
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq221_HTML.gif  :  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq222_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ27_HTML.gif
        (2.19)

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

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

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

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

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

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

        where http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq243_HTML.gif , then http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq246_HTML.gif and a negative solution http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq247_HTML.gif with http://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 http://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, http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq252_HTML.gif and http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ32_HTML.gif
        (3.1)
        and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq254_HTML.gif . The functionals http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq255_HTML.gif are defined as
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq256_HTML.gif and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq258_HTML.gif is a critical point of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq259_HTML.gif (or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq260_HTML.gif ), then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq261_HTML.gif (or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq262_HTML.gif ).

        Lemma 3.2.

        The functionals http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq264_HTML.gif in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq265_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq266_HTML.gif is bounded, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq267_HTML.gif as http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq269_HTML.gif . The case of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq270_HTML.gif is completely similar. Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq272_HTML.gif is bounded. Suppose that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq274_HTML.gif and for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq275_HTML.gif , either http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq276_HTML.gif or http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq277_HTML.gif is bounded.

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

        Because http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq289_HTML.gif , we see that if http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq291_HTML.gif is positive. Since this contradicts http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq292_HTML.gif , we conclude that http://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 http://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, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq296_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq299_HTML.gif with http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq301_HTML.gif . Firstly, we will prove that http://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, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq304_HTML.gif . In fact, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq305_HTML.gif , and http://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, http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq309_HTML.gif , then for any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq310_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq311_HTML.gif , such that
        http://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,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ41_HTML.gif
        (3.10)
        Take http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq312_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq313_HTML.gif . If we set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq314_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ42_HTML.gif
        (3.11)
        Since for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq315_HTML.gif , if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq316_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq317_HTML.gif for every http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq318_HTML.gif and hence
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ43_HTML.gif
        (3.12)
        where http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq319_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq320_HTML.gif . If we take
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ44_HTML.gif
        (3.13)

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

        So, if we take http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq327_HTML.gif sufficiently large such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq328_HTML.gif and for http://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, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq331_HTML.gif . And for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq332_HTML.gif , we claim that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq333_HTML.gif . If not, set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq334_HTML.gif , then for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq335_HTML.gif . By http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq336_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq337_HTML.gif . Hence, http://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 http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ47_HTML.gif
        (3.16)
        and that there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq340_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ48_HTML.gif
        (3.17)
        This implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq341_HTML.gif satisfies
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ49_HTML.gif
        (3.18)
        Hence, the eigenvalue problem
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ50_HTML.gif
        (3.19)
        has an eigenvalue http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq342_HTML.gif . Condition http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq344_HTML.gif . Since http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq346_HTML.gif must be finite and http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq348_HTML.gif . Finally, choose the neighborhood http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq349_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq350_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq351_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq352_HTML.gif , then
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq353_HTML.gif , one has
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq354_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq355_HTML.gif ,
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq356_HTML.gif and http://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
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq359_HTML.gif satisfies (2.11). In fact, let
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ55_HTML.gif
        (3.24)
        By http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq360_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq361_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq362_HTML.gif , we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ56_HTML.gif
        (3.25)

        Hence, if we set http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq364_HTML.gif implies that there exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq365_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq366_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq367_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ57_HTML.gif
        (3.26)
        Hence, we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ58_HTML.gif
        (3.27)
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq368_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq369_HTML.gif satisfying
        http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq370_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq371_HTML.gif , and http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq373_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq374_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq375_HTML.gif are the only critical points of http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ61_HTML.gif
        (3.30)

        This is impossible. Thus, http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq379_HTML.gif and http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq383_HTML.gif , then we have
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ62_HTML.gif
        (3.31)
        Hence, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq384_HTML.gif may coincide with http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq385_HTML.gif or http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq387_HTML.gif , then
        http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq389_HTML.gif large enough, such that
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq390_HTML.gif . For any http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq391_HTML.gif , define http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq392_HTML.gif as
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ65_HTML.gif
        (3.34)
        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq393_HTML.gif . The functional http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq394_HTML.gif  :  http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq395_HTML.gif is defined as
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ66_HTML.gif
        (3.35)
        It is obvious that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq396_HTML.gif is of class http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ67_HTML.gif
        (3.36)

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

        We claim that if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq404_HTML.gif is a ball in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq405_HTML.gif containing zero, then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq406_HTML.gif . In fact, since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq407_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq408_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq409_HTML.gif . Hence, for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq410_HTML.gif , we have
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq411_HTML.gif is positive on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq412_HTML.gif . Then, for each http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq413_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq414_HTML.gif , we have
        http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ70_HTML.gif
        (3.39)

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

        Now, let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq416_HTML.gif . We claim that for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq417_HTML.gif large enough and for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq418_HTML.gif , the function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq419_HTML.gif has no zero on http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq421_HTML.gif and for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq422_HTML.gif , we have
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq423_HTML.gif , for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq424_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq425_HTML.gif large enough such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq426_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq427_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq428_HTML.gif , for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq429_HTML.gif , take http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq430_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ72_HTML.gif
        (3.41)
        For http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq431_HTML.gif , take http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq432_HTML.gif , then
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ73_HTML.gif
        (3.42)
        Hence, if we take http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq433_HTML.gif , then for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq434_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq435_HTML.gif , and for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq436_HTML.gif , we have http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq437_HTML.gif . So, if we let
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ74_HTML.gif
        (3.43)
        then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq438_HTML.gif for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq439_HTML.gif . And for all http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq440_HTML.gif , we have
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq441_HTML.gif large enough, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq442_HTML.gif has no zero point on http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq443_HTML.gif for each http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq446_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq447_HTML.gif ) is a neighborhood of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq448_HTML.gif ( http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq449_HTML.gif ) containing no other critical points, then

        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ77_HTML.gif
        (3.46)
          Assume that http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ78_HTML.gif
        (3.47)
        Hence, if http://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
        http://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 http://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
        http://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 http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq456_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ81_HTML.gif
        (3.50)
        Moreover, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq458_HTML.gif is the unique member of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq459_HTML.gif such that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ82_HTML.gif
        (3.51)
        The function http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq460_HTML.gif defined by http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq462_HTML.gif is of class http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq463_HTML.gif . Because http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq464_HTML.gif , (3.27) implies that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq465_HTML.gif as http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq466_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq467_HTML.gif , there must exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq468_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq469_HTML.gif . Take http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq470_HTML.gif , then http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq472_HTML.gif is a neighborhood of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq473_HTML.gif containing no other critical points of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq474_HTML.gif , taking http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq475_HTML.gif , then http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ83_HTML.gif
        (3.52)

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

        Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq478_HTML.gif be large enough so that if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq479_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq480_HTML.gif . Because http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq481_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq482_HTML.gif is of class http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq483_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq484_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq485_HTML.gif for http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq486_HTML.gif . Because http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq487_HTML.gif is coercive, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq488_HTML.gif . Hence, if we set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq489_HTML.gif  :  http://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
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ84_HTML.gif
        (3.53)
        Suppose that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq491_HTML.gif is finite. Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq492_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq493_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq494_HTML.gif be disjoint open bounded regions in http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq495_HTML.gif such that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq497_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq499_HTML.gif . So far, we have proved that
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ85_HTML.gif
        (3.54)
        (i) If http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq500_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq501_HTML.gif is sign changing. Let http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq503_HTML.gif such that http://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
        http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq505_HTML.gif such that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq507_HTML.gif and http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq510_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq511_HTML.gif be a neighborhood of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq512_HTML.gif such that http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq514_HTML.gif such that if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq515_HTML.gif is a neighborhood of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq516_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq517_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq518_HTML.gif . Thus,
          http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq519_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq520_HTML.gif . Finally,
        http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_Equ88_HTML.gif
        (3.57)

        Thus, there must exist http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq521_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq522_HTML.gif . Thus, the set http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq523_HTML.gif together with a critical point http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq524_HTML.gif of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq525_HTML.gif in http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq527_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq528_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq529_HTML.gif is a sign-changing solution, and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq530_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq531_HTML.gif , and http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq533_HTML.gif is even.

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

          Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq535_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq536_HTML.gif , and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq537_HTML.gif be as above. If http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq539_HTML.gif , hence http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq540_HTML.gif . Because http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq541_HTML.gif , there exists http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq542_HTML.gif such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq543_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq544_HTML.gif . So, if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq545_HTML.gif , then http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq546_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq547_HTML.gif is a local maximum of http://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, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq549_HTML.gif is a strictly local maximum of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq550_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq551_HTML.gif be such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq552_HTML.gif if http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq553_HTML.gif . Since http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq554_HTML.gif , http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq555_HTML.gif is path connected. Thus, http://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, http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq558_HTML.gif . By (v) of Lemma 2.10, http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq559_HTML.gif , and hence http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq560_HTML.gif . Let http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq561_HTML.gif be neighborhoods of http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq562_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq563_HTML.gif , respectively, such that http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq564_HTML.gif and http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq565_HTML.gif . Thus,
          http://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 http://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 http://static-content.springer.com/image/art%3A10.1155%2F2011%2F172818/MediaObjects/13661_2010_Article_26_IEq567_HTML.gif and that http://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

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