International Journal of Mathematics and Mathematical Sciences

International Journal of Mathematics and Mathematical Sciences / 2010 / Article

Research Article | Open Access

Volume 2010 |Article ID 907453 | 21 pages | https://doi.org/10.1155/2010/907453

Existence of Multiple Solutions of a Second-Order Difference Boundary Value Problem

Academic Editor: Martin Bohner
Received03 Jul 2009
Revised05 Feb 2010
Accepted07 Mar 2010
Published22 Mar 2010

Abstract

This paper studies the existence of multiple solutions of the second-order difference boundary value problem Ξ”2𝑒(π‘›βˆ’1)+π‘‰ξ…ž(𝑒(𝑛))=0, π‘›βˆˆβ„€(1,𝑇), 𝑒(0)=0=𝑒(𝑇+1). By applying Morse theory, critical groups, and the mountain pass theorem, we prove that the previous equation has at least three nontrivial solutions when the problem is resonant at the eigenvalue πœ†π‘˜(π‘˜β‰₯2) of linear difference problem Ξ”2𝑒(π‘›βˆ’1)+πœ†π‘’(𝑛)=0, π‘›βˆˆβ„€(1,𝑇), 𝑒(0)=0=𝑒(𝑇+1) near infinity and the trivial solution of the first equation is a local minimizer under some assumptions on 𝑉.

1. Introduction

Let ℝ, β„•, and β„€ be the sets of real numbers, natural numbers, and integers, respectively. For any π‘Ž,π‘βˆˆβ„€, π‘Žβ‰€π‘, define β„€(π‘Ž,𝑏)={π‘Ž,π‘Ž+1,…,𝑏}.

Consider the second-order difference boundary value problem (BVP)

Ξ”2𝑒(π‘›βˆ’1)+π‘‰ξ…ž(𝑒(𝑛))=0,π‘›βˆˆβ„€(1,𝑇),𝑒(0)=0=𝑒(𝑇+1),(1.1) where π‘‰βˆˆπΆ2(ℝ,ℝ) and Ξ” denotes the forward difference operator defined by Δ𝑒(𝑛)=𝑒(𝑛+1)βˆ’π‘’(𝑛), Ξ”2𝑒(𝑛)=Ξ”(Δ𝑒(𝑛)).

By a solution 𝑒 of the BVP (1.1), we mean a real sequence {𝑒(𝑛)}𝑇+1𝑛=0(=(𝑒(0),𝑒(1),…,𝑒(𝑇+1))) satisfying the BVP (1.1). For 𝑒={𝑒(𝑛)}𝑇+1𝑛=0 with 𝑒(0)=0=𝑒(𝑇+1), we say that 𝑒≠0 if there exists at least one π‘›βˆˆβ„€(1,𝑇) such that 𝑒(𝑛)β‰ 0. We say that 𝑒 is positive (and write 𝑒>0) if for all π‘›βˆˆβ„€(1,𝑇), 𝑒(𝑛)>0, and similarly, 𝑒 is negative (𝑒<0) if for all π‘›βˆˆβ„€(1,𝑇), 𝑒(𝑛)<0. The aim of this paper is to obtain the existence of multiple solutions of the BVP (1.1) and analyse the sign of solutions.

Recently, a few authors applied the minimax methods to examine the difference boundary value problems. For example, in [1], Agarwal et al. employed the Mountain Pass Lemma to study the following BVP:

Ξ”2𝑒(π‘›βˆ’1)+𝑓(𝑛,𝑒(𝑛))=0,π‘›βˆˆβ„€(1,𝑇),𝑒(0)=0=𝑒(𝑇+1)(1.2) and obtained the existence of multiple positive solutions, where 𝑓 may be singular at 𝑒=0. In [2], Jiang and Zhou employed the Mountain Pass Lemma together with strongly monotone operator principle, to study the following difference BVP:

Ξ”2𝑒(π‘›βˆ’1)+𝑓(𝑛,𝑒(𝑛))=0,π‘›βˆˆβ„€(1,𝑇),𝑒(0)=0=Δ𝑒(𝑇)(1.3) and obtained existence and uniqueness results, where π‘“βˆΆβ„€(1,𝑇)×ℝ→ℝ is continuous. In [3], Cai and Yu employed the Linking Theorem and the Mountain Pass Lemma to study the following difference BVP:

Δ𝑝(𝑛)(Δ𝑒(π‘›βˆ’1))𝛿+π‘ž(𝑛)𝑒𝛿(𝑛)=𝑓(𝑛,𝑒(𝑛)),π‘›βˆˆβ„€(1,𝑇),Δ𝑒(0)=𝐴,𝑒(𝑇+1)=𝐡(1.4) and obtained the existence of multiple solutions, where 𝛿>0 is the ratio of odd positive integers, {𝑝(𝑛)}𝑇+1𝑛=1 and {π‘ž(𝑛)}𝑇𝑛=1 are real sequences, 𝑝(𝑛)β‰ 0 for all π‘›βˆˆβ„€(1,𝑇+1), and 𝐴, 𝐡 are two given constants, π‘“βˆΆβ„€(1,𝑇)×ℝ→ℝ is continuous.

Although applications of the minimax methods in the field of the difference BVP have attracted some scholarly attention in the recent years, efforts in applying Morse theory to the difference BVP are scarce. The main purpose of this paper is to develop a new approach to the BVP (1.1) by using Morse theory. To this end, we first consider the following linear difference eigenvalue problem:

Ξ”2𝑒(π‘›βˆ’1)+πœ†π‘’(𝑛)=0,π‘›βˆˆβ„€(1,𝑇),𝑒(0)=0=𝑒(𝑇+1).(1.5)

On the above eigenvalue problem, the following results hold; see [4].

Proposition 1.1. The eigenvalues of (1.5) are πœ†=πœ†π‘™=4sin2π‘™πœ‹2(𝑇+1),𝑙=1,2,…,𝑇,(1.6) and the corresponding eigenfunction with πœ†π‘™ is πœ™π‘™(𝑛)=sin(π‘™πœ‹π‘›/(𝑇+1)), 𝑙=1,2,…,𝑇.

Remark 1.2. (1) The set of functions {πœ™π‘™(𝑛),𝑙=1,2,…,𝑇} is orthogonal on β„€(1,𝑇) with respect to the weight function π‘Ÿ(𝑛)≑1, that is, 𝑇𝑛=1ξ€·πœ™π‘™(𝑛),πœ™π‘—ξ€Έ(𝑛)=0βˆ€π‘™β‰ π‘—.(1.7) Moreover, for each π‘™βˆˆβ„€(1,𝑇), βˆ‘π‘‡π‘›=1sin2(π‘™πœ‹π‘›/(𝑇+1))=(𝑇+1)/2.
(2) It is easy to see that πœ™1 is positive and πœ™π‘™ changes sign for each π‘™βˆˆβ„€(2,𝑇), that is, {π‘›βˆΆπœ™π‘™(𝑛)>0}β‰ βˆ… and {π‘›βˆΆπœ™π‘™(𝑛)<0}β‰ βˆ….

For (1.1), we assume that

𝑉(0)=π‘‰ξ…žπ‘‰(0)=0,(1.8)ξ…žξ…ž(∞)∢=lim|𝑑|β†’βˆžπ‘‰ξ…ž(𝑑)𝑑=πœ†π‘˜,(1.9) where πœ†π‘˜ is an eigenvalue of (1.5). Hence the BVP (1.1) has a trivial solution 𝑒≑0. And we say that BVP (1.1) is resonant at infinity if (1.9) holds.

Let

π‘Šβˆ’ξ€½πœ™=span1,πœ™2,…,πœ™π‘˜βˆ’1ξ€Ύ,π‘Š0ξ€½πœ™=spanπ‘˜ξ€Ύ,π‘Š+ξ€½πœ™=spanπ‘˜+1,πœ™π‘˜+2,…,πœ™π‘‡ξ€Ύ.(1.10) Let ∫𝐺(𝑑)=𝑑0πΊξ…ž(𝑠)𝑑𝑠=𝑉(𝑑)βˆ’(πœ†π‘˜/2)𝑑2. By (1.9) we have

lim|𝑑|β†’βˆžπΊξ…ž(𝑑)𝑑=0.(1.11) Assume that the following conditions on πΊξ…ž(𝑑) hold.

(𝐺±) If β€–π‘’π‘šβ€–β†’βˆž such that β€–π‘£π‘šβ€–/β€–π‘’π‘šβ€–β†’1, then there exist 𝛿>0 and π‘€βˆˆβ„• such that

±𝑇𝑛=1ξ€·πΊξ…žξ€·π‘’π‘šξ€Έ(𝑛),π‘£π‘šξ€Έ(𝑛)β‰₯𝛿,βˆ€π‘šβ‰₯𝑀,(1.12) where π‘’π‘š=π‘£π‘š+π‘€π‘š, π‘£π‘šβˆˆπ‘Š0, π‘€π‘šβˆˆπ‘ŠβˆΆ=π‘Š+βŠ•π‘Šβˆ’.

The main result of this paper is as follows.

Theorem 1.3. Let (1.8), (1.9) hold and (𝑉1)π‘‰ξ…žξ…ž(𝑑)>0 for all π‘‘βˆˆβ„, (𝑉2)π‘‰ξ…žξ…ž(0)<πœ†1hold. Then the BVP (1.1) has at least three nontrivial solutions, with one positive solution and one negative solution, in each of the following cases: (i)(G+) and π‘˜β‰₯2; (ii)(Gβˆ’) and π‘˜β‰₯3.

To the author’s best knowledge, only Bin et al. [5] deal with the existence and multiplicity of nontrivial periodic solutions for asymptotically linear resonant difference problem by the aid of Su [6]. In [5], 𝐺 satisfies

||πΊξ…ž||(𝑧)≀𝑐1|𝑧|𝑠+𝑐2,(1.13)limβ€–π‘£β€–β†’βˆžinfπ‘£βˆˆπ‘Š01‖𝑣‖2𝑠𝐺(𝑧)β‰₯4𝛽2,𝛿𝑇(1.14) where 𝑐1>0, 𝑐2>0, π‘ βˆˆ(0,1),𝛽=𝑐1𝑇(1βˆ’π‘ )/2, 𝛿>0. In [5], the authors obtained the existence of one nontrivial periodic solution. Notice that (1.13) implies that (1.11) holds; however, (𝐺±) is not covered by (1.14). In fact, conditions (1.13) and (1.14) are borrowed from [6]. The conditions in Theorem 1.3 coincide with the assumptions of Theorem 1 in [7]. The aim of this paper is to develop a new approach to study the discrete systems by using Morse theory, minimax theorems, and some analysis technique. We wish to have some breakthrough points with the aid of the method of discretization.

The remaining part of this paper proceeds as follows. In the next section, we establish the variational framework of the BVP (1.1) and collect some results which will be used in the proof of Theorem 1.3. In Section 3, we give the proof of Theorem 1.3. Finally, in Section 4, we give an example to illustrate our main result and summarize conclusions and future directions.

2. Variational Framework and Auxiliary Results

Let

𝐸=π‘’βˆΆπ‘’={𝑒(𝑛)}𝑇+1𝑛=0ξ€Ύ.with𝑒(0)=0=𝑒(𝑇+1)βˆˆβ„(2.1)𝐸 can be equipped with the norm β€–β‹…β€– and the inner product βŸ¨β‹…,β‹…βŸ© as follows:

‖𝑒‖=𝑇𝑛=0||||Δ𝑒(𝑛)2ξƒͺ0π‘₯0200𝑑1/2,βˆ€π‘’βˆˆπΈ,βŸ¨π‘’,π‘£βŸ©=𝑇𝑛=0(Δ𝑒(𝑛),Δ𝑣(𝑛)),βˆ€π‘’,π‘£βˆˆπΈ,(2.2) where |β‹…| denotes the Euclidean norm in ℝ and (β‹…,β‹…) denotes the usual scalar product in ℝ. It is easy to see that (𝐸,βŸ¨β‹…,β‹…βŸ©) is a Hilbert space. Consider the functional defined on 𝐸 by

1𝐽(𝑒)=2𝑇𝑛=0|Δ𝑒(𝑛)|2βˆ’π‘‡ξ“π‘›=1𝑉(𝑒(𝑛)).(2.3) We claim that if π‘’βˆˆπΈ is a critical point of 𝐽, then 𝑒 is precisely a solution of the BVP (1.1). Indeed, for every 𝑒,π‘£βˆˆπΈ, we have

βŸ¨π½ξ…ž(𝑒),π‘£βŸ©=𝑇𝑛=0(Δ𝑒(𝑛),Δ𝑣(𝑛))βˆ’π‘‡ξ“π‘›=1ξ€·π‘‰ξ…žξ€Έ(𝑒(𝑛)),𝑣(𝑛)=βˆ’π‘‡ξ“π‘›=1ξ€·Ξ”2𝑒(π‘›βˆ’1)+π‘‰ξ…žξ€Έ.(𝑒(𝑛)),𝑣(𝑛)(2.4)

So, if π½ξ…ž(𝑒)=0, then we have

𝑇𝑛=1ξ€·Ξ”2𝑒(π‘›βˆ’1)+π‘‰ξ…žξ€Έ(𝑒(𝑛)),𝑣(𝑛)=0.(2.5) Since π‘£βˆˆπΈ is arbitrary, we obtain

Ξ”2𝑒(π‘›βˆ’1)+π‘‰ξ…ž(𝑒(𝑛))=0,π‘›βˆˆβ„€(1,𝑇).(2.6) Therefore, we reduce the problem of finding solutions of the BVP (1.1) to that of seeking critical points of the functional 𝐽 in 𝐸.

According to Proposition 1.1 and Remark 1.2, 𝐸 can be decomposed as 𝐸=π‘Šβˆ’βŠ•π‘Š0βŠ•π‘Š+. For all π‘’βˆˆπΈ, denote 𝑒=𝑀0+𝑀++π‘€βˆ’ with 𝑀0βˆˆπ‘Š0, 𝑀+βˆˆπ‘Š+, and π‘€βˆ’βˆˆπ‘Šβˆ’, then we have the following Wirtinger type inequalities:

πœ†1𝑇𝑛=1(𝑒(𝑛),𝑒(𝑛))≀‖𝑒‖2β‰€πœ†π‘‡π‘‡ξ“π‘›=1πœ†(𝑒(𝑛),𝑒(𝑛)),βˆ€π‘’βˆˆπΈ,(2.7)1𝑇𝑛=1(π‘€βˆ’(𝑛),π‘€βˆ’(𝑛))β‰€β€–π‘€βˆ’β€–2β‰€πœ†π‘‡π‘˜βˆ’1𝑛=1(π‘€βˆ’(𝑛),π‘€βˆ’(𝑛)),βˆ€π‘€βˆ’βˆˆπ‘Šβˆ’πœ†,(2.8)π‘‡π‘˜+1𝑛=1𝑀+(𝑛),𝑀+≀‖‖𝑀(𝑛)+β€–β€–2β‰€πœ†π‘‡π‘‡ξ“π‘›=1𝑀+(𝑛),𝑀+ξ€Έ(𝑛),βˆ€π‘€+βˆˆπ‘Š+,(2.9) see [4] for details.

Now we collect some results on Morse theory and the minimax methods.

Let 𝐸 be a real Hilbert space and 𝐽∈𝐢1(𝐸,ℝ). Denote

𝐽𝑐={π‘’βˆˆπΈβˆΆπ½(𝑒)≀𝑐},𝒦𝑐={π‘’βˆˆπΈβˆΆπ½β€²(𝑒)=0,𝐽(𝑒)=𝑐}(2.10) for π‘βˆˆβ„. The following is the definition of the Palais-Smale condition ((PS) condition).

Definition 2.1. The functional 𝐽 satisfies the (PS) condition if any sequence {π‘’π‘š}βŠ‚πΈ such that {𝐽(π‘’π‘š)} is bounded and 𝐽′(π‘’π‘š)β†’0 as π‘šβ†’βˆž has a convergent subsequence.

In [8], Cerami introduced a weak version of the (PS) condition as follows.

Definition 2.2. The functional 𝐽 satisfies the Cerami condition ((𝐢) condition) if any sequence {π‘’π‘š}βŠ‚πΈ such that {𝐽(π‘’π‘š)} is bounded and (1+β€–π‘’π‘šβ€–)β€–π½ξ…ž(π‘’π‘š)β€–β†’0 as π‘šβ†’βˆž has a convergent subsequence.

If 𝐽 satisfies the (PS) condition or the (𝐢) condition, then 𝐽 satisfies the following deformation condition which is essential in Morse theory (cf. [9, 10]).

Definition 2.3. The functional 𝐽 satisfies the (𝐷𝑐) condition at the level π‘βˆˆβ„ if for any πœ–>0 and any neighborhood 𝒩 of 𝒦𝑐, there are πœ–>0 and a continuous deformation πœ‚βˆΆ[0,1]×𝐸→𝐸 such that(1)πœ‚(0,𝑒)=𝑒 for all π‘’βˆˆπΈ;(2)πœ‚(𝑑,𝑒)=𝑒 for all π‘’βˆ‰π½βˆ’1([π‘βˆ’πœ–,𝑐+πœ–]);(3)𝐽(πœ‚(𝑑,𝑒)) is nonincreasing in 𝑑 for any π‘’βˆˆπΈ;(4)πœ‚(1,𝐽𝑐+πœ–β§΅π’©)βŠ‚π½π‘βˆ’πœ–. 𝐽 satisfies the (𝐷) condition if 𝐽 satisfies the (𝐷𝑐) condition for all π‘βˆˆβ„.

Let 𝑒0 be an isolated critical point of 𝐽 with 𝐽(𝑒0)=π‘βˆˆβ„, and let π‘ˆ be a neighborhood of 𝑒0, the group

πΆπ‘žξ€·π½,𝑒0ξ€ΈβˆΆ=π»π‘žξ€·π½π‘βˆ©π‘ˆ,π½π‘ξ€½π‘’βˆ©π‘ˆβ§΅0ξ€Ύξ€Έ,π‘žβˆˆβ„€,(2.11) is called the π‘žth critical group of 𝐽 at 𝑒0, where π»π‘ž(𝐴,𝐡) denotes the π‘žth singular relative homology group of the pair (𝐴,𝐡) over a field 𝐹, which is defined to be quotient π»π‘ž(𝐴,𝐡)=π‘π‘ž(𝐴,𝐡)/π΅π‘ž(𝐴,𝐡), where π‘π‘ž(𝐴,𝐡) is the π‘žth singular relative closed chain group and π΅π‘ž(𝐴,𝐡) is the π‘žth singular relative boundary chain group.

Let 𝒦={π‘’βˆˆπΈβˆΆπ½β€²(𝑒)=0}. If 𝐽(𝒦) is bounded from below by π‘Žβˆˆβ„ and 𝐽 satisfies the (𝐷𝑐) condition for all π‘β‰€π‘Ž, then the group

πΆπ‘ž(𝐽,∞)∢=π»π‘ž(𝐸,π½π‘Ž),π‘žβˆˆβ„€,(2.12) is called the π‘žth critical group of 𝐽 at infinity [11].

Assume that #𝒦<∞ and 𝐽 satisfies the (𝐷) condition. The Morse-type numbers of the pair (𝐸,π½π‘Ž) are defined by

π‘€π‘ž=π‘€π‘ž(𝐸,π½π‘Ž)=ξ“π‘’βˆˆπ’¦dimπΆπ‘ž(𝐽,𝑒),(2.13) and the Betti numbers of the pair (𝐸,π½π‘Ž) are

π›½π‘žβˆΆ=dimπΆπ‘ž(𝐽,∞).(2.14) By Morse theory [12, 13], the following relations hold:

π‘žξ“π‘—=0(βˆ’1)π‘žβˆ’π‘—π‘€π‘—β‰₯π‘žξ“π‘—=0(βˆ’1)π‘žβˆ’π‘—π›½π‘—,π‘žβˆˆβ„€,βˆžξ“π‘ž=0π‘€π‘ž=βˆžξ“π‘ž=0π›½π‘ž.(2.15)

Thus, if πΆπ‘ž(𝐽,∞)≇0, for some π‘˜βˆˆβ„€, then there must exist a critical point 𝑒 of 𝐽 with πΆπ‘ž(𝐽,𝑒)≇0, which can be rephrased as follows.

Proposition 2.4. Let 𝐸 be a real Hilbert space and 𝐽∈𝐢2(𝐸,ℝ). Assume that #𝒦<∞ and that 𝐽 satisfies the (𝐷) condition. If there exists some π‘žβˆˆβ„€ such that πΆπ‘ž(𝐽,∞)≇0, then 𝐽 must have a critical point 𝑒 with πΆπ‘ž(𝐽,𝑒)≇0.

In order to prove our main result, we need the following result about the critical group on πΆπ‘ž(𝐽,∞).

Proposition 2.5. Let the functional π½βˆΆπΈβ†’β„ be of the form 1𝐽(𝑒)=2βŸ¨π΄π‘’,π‘’βŸ©+𝑄(𝑒),(2.16) where π΄βˆΆπΈβ†’πΈ is a self-adjoint linear operator such that 0 is isolated in 𝜎(𝐴), the spectrum of 𝐴. Assume that π‘„βˆˆπΆ1(𝐸,ℝ) satisfies limβ€–π‘’β€–β†’βˆžβ€–β€–π‘„ξ…žβ€–β€–(𝑒)‖𝑒‖=0.(2.17) Denote π‘‰βˆΆ=ker𝐴, π‘ŠβˆΆ=π‘‰βŸ‚=π‘Š+βŠ•π‘Šβˆ’, where π‘Š+ (π‘Šβˆ’) is the subspace of 𝐸 on which 𝐴 is positive (negative) definite. Assume that πœ‡βˆΆ=dimπ‘Šβˆ’, 𝜈∢=dim𝑉≠0 are finite and that 𝐽 satisfies the (𝐷) condition. Then πΆπ‘ž(𝐽,∞)β‰…π›Ώπ‘ž,π‘˜Β±πΉ,π‘žβˆˆβ„€,(2.18) provided that 𝐽 satisfies the angle conditions at infinity. (𝐴𝐢±∞): there exist 𝑀>0 and π›Όβˆˆ(0,1) such that Β±βŸ¨π½β€²(𝑒),π‘£βŸ©β‰₯0for𝑒=𝑣+𝑀,‖𝑒‖β‰₯𝑀,‖𝑀‖≀𝛼‖𝑒‖,(2.19) where π‘˜+=πœ‡, π‘˜βˆ’=πœ‡+𝜈, π‘£βˆˆπ‘‰, and π‘€βˆˆπ‘Š.

Remark 2.6. Conditions (2.16) and (2.17) imply that 𝐽 is asymptotically quadratic. Bartsch and Li [11] introduced the notion of critical groups at infinity and proved that if 𝐽 satisfied some angle properties at infinity, the critical groups can be completely figured out. Proposition 2.5 is a slight improvement of [11, Proposition 3.10] by Su and Zhao [7]. There are many other papers considering concrete problems by computing the critical groups at infinity with different methods, for example, see [14–17].

We will use the Mountain Pass Lemma (cf. [12, 18]) in our proof.

Let 𝐡𝜌 denote the open ball in 𝐸 about 0 of radius π‘Ÿ and let πœ•π΅πœŒ denote its boundary.

Theorem 2.7 (mountain pass lemma). Let 𝐸 be a real Banach space and 𝐽∈𝐢1(𝐸,ℝ) satisfying the (PS) condition. Suppose 𝐽(0)=0 and that (𝐽1) there are constants 𝜌>0,π‘Ž>0 such that 𝐽|πœ•π΅πœŒβ‰₯π‘Ž>0, (𝐽2) there is a 𝑒0∈𝐸⧡𝐡𝜌 such that 𝐽(𝑒0)≀0,then 𝐽 possesses a critical value 𝑐β‰₯π‘Ž. Moreover 𝑐 can be characterized as 𝑐=infβ„ŽβˆˆΞ“supπ‘ βˆˆ[0,1]𝐽(β„Ž(𝑠)),(2.20) where ξ€½([]Ξ“=β„ŽβˆˆπΆ0,1,𝐸)βˆ£β„Ž(0)=0,β„Ž(1)=𝑒0ξ€Ύ.(2.21)

Definition 2.8 (mountain pass point). An isolated critical point 𝑒 of 𝐽 is called a mountain pass point, if 𝐢1(𝐽,𝑒)≇0.

The following result is useful in computing the critical group of a mountain pass point; see [13, 19] for details.

Theorem 2.9. Let 𝐸 be a real Hilbert space. Suppose that 𝐽∈𝐢2(𝐸,ℝ) has a mountain pass point 𝑒, and that π½ξ…žξ…ž(𝑒) is a Fredholm operator with finite Morse index, satisfying π½ξ…žξ…žξ€·π‘’0𝐽β‰₯0,0βˆˆπœŽξ…žξ…žξ€·π‘’0ξ€·π½ξ€Έξ€ΈβŸΉdimkerξ…žξ…žξ€·π‘’0ξ€Έξ€Έ=1,(2.22) then πΆπ‘žξ€·π½,𝑒0ξ€Έβ‰…π›Ώπ‘ž,1𝐹,π‘žβˆˆβ„€.(2.23)

3. Proof of Theorem 1.3

We give the proof of Theorem 1.3 in this section. Firstly, we prove that the functional 𝐽 satisfies the (𝐢) condition (Lemma 3.1) and compute the critical group πΆπ‘ž(𝐽,∞) (Lemma 3.2). Then, we employ the cut-off technique and the Mountain Pass Lemma to obtain two critical points 𝑒+,π‘’βˆ’ of 𝐽 and compute the critical groups πΆπ‘ž(𝐽,𝑒+) and πΆπ‘ž(𝐽,π‘’βˆ’) (Lemmas 3.3 and 3.4). Finally, we prove Theorem 1.3.

Rewrite the functional 𝐽 as

1𝐽(𝑒)=2𝑇𝑛=0|Δ𝑒(𝑛)|2βˆ’πœ†π‘˜2𝑇𝑛=1|𝑒(𝑛)|2βˆ’π‘‡ξ“π‘›=1𝐺(𝑒(𝑛)),βˆ€π‘’βˆˆπΈ.(3.1)

Lemma 3.1. Let (1.8) and (1.9) hold. If 𝐺 satisfies (𝐺±), then the functional 𝐽 satisfies the (𝐢) condition.

Proof. We only prove the case where (𝐺+) holds. Let {π‘’π‘š}βŠ‚πΈ such that π½ξ€·π‘’π‘šξ€Έξ€·β€–β€–π‘’βŸΆπ‘βˆˆβ„,1+π‘šβ€–β€–ξ€Έβ€–β€–π½ξ…žξ€·π‘’π‘šξ€Έβ€–β€–βŸΆ0asπ‘šβŸΆβˆž.(3.2) Then for all πœ‘βˆˆπΈ, we have βŸ¨π½ξ…žξ€·π‘’π‘šξ€Έ,πœ‘βŸ©=βŸ¨π‘’π‘š,πœ‘βŸ©βˆ’πœ†π‘˜π‘‡ξ“π‘›=1ξ€·π‘’π‘šξ€Έβˆ’(𝑛),πœ‘(𝑛)𝑇𝑛=1ξ€·πΊξ…žξ€·π‘’π‘šξ€Έξ€Έ.(𝑛),πœ‘(𝑛)(3.3) Denote π‘’π‘š=π‘£π‘š+𝑀+π‘š+π‘€βˆ’π‘š with π‘£π‘šβˆˆπ‘Š0, 𝑀+π‘šβˆˆπ‘Š+ and π‘€βˆ’π‘šβˆˆπ‘Šβˆ’. Since 𝐸 is a finite-dimensional Hilbert space, it suffices to show that {π‘’π‘š} is bounded. Suppose that {π‘’π‘š} is unbounded. Passing to a subsequence we may assume that β€–π‘’π‘šβ€–β†’βˆž as π‘šβ†’βˆž.
By (1.11), for any πœ–>0, there exists π‘βˆˆβ„ such that ||πΊξ…ž||(𝑑)β‰€πœ–|𝑑|+𝑏,βˆ€π‘‘βˆˆβ„.(3.4) Let πœ‘=𝑀+π‘š in (3.3). Then by (2.7), (2.9), and (3.4), we have 𝑐1‖‖𝑀+π‘šβ€–β€–2ξ‚΅πœ†βˆΆ=1βˆ’π‘˜πœ†π‘˜+1‖‖𝑀+π‘šβ€–β€–2≀‖‖𝑀+π‘šβ€–β€–2βˆ’πœ†π‘˜π‘‡ξ“π‘›=1𝑀+π‘š(𝑛),𝑀+π‘šξ€Έ=(𝑛)𝑇𝑛=1ξ€·ξ€·π‘’πΊβ€²π‘š(𝑛),𝑀+π‘š(𝑒𝑛)+βŸ¨π½β€²π‘šξ€Έ,𝑀+π‘šβŸ©β‰€β€–β€–π‘€+π‘šβ€–β€–+𝑇𝑛=1ξ€·πœ–||π‘’π‘š||ξ€Έ||𝑀(𝑛)+𝑏+π‘š||≀‖‖𝑀(𝑛)+π‘šβ€–β€–+πœ–βˆšπœ†1πœ†π‘˜+1β€–β€–π‘’π‘šβ€–β€–β€–β€–π‘€+π‘šβ€–β€–+π‘βˆšπ‘‡βˆšπœ†π‘˜+1‖‖𝑀+π‘šβ€–β€–βˆΆ=𝑐2‖‖𝑀+π‘šβ€–β€–+𝑐3β€–β€–π‘’π‘šβ€–β€–β€–β€–π‘€+π‘šβ€–β€–,(3.5) where 𝑐1πœ†=1βˆ’π‘˜πœ†π‘˜+1>0,𝑐2π‘βˆš=1+π‘‡βˆšπœ†π‘˜+1,𝑐3=πœ–βˆšπœ†1πœ†π‘˜+1.(3.6) And since πœ–>0 is arbitrary, we have ‖‖𝑀+π‘šβ€–β€–β€–β€–π‘’π‘šβ€–β€–βŸΆ0asπ‘šβŸΆβˆž.(3.7) Similarly, let πœ‘=π‘€βˆ’π‘š in (3.3), by (2.8) and (3.4) we get βˆ’π‘4β€–β€–π‘€βˆ’π‘šβ€–β€–2ξ‚΅πœ†=∢1βˆ’π‘˜πœ†π‘˜βˆ’1ξ‚Άβ€–β€–π‘€βˆ’π‘šβ€–β€–2β‰₯β€–β€–π‘€βˆ’π‘šβ€–β€–2βˆ’πœ†π‘˜π‘‡ξ“π‘›=1ξ€·π‘€βˆ’π‘š(𝑛),π‘€βˆ’π‘šξ€Έ=(𝑛)𝑇𝑛=1ξ€·πΊξ…žξ€·π‘’π‘š(𝑛),π‘€βˆ’π‘š(𝑛)+βŸ¨π½ξ…žξ€·π‘’π‘šξ€Έ,π‘€βˆ’π‘šβŸ©β€–β€–π‘€β‰₯βˆ’βˆ’π‘šβ€–β€–βˆ’π‘‡ξ“π‘›=1ξ€·πœ–||π‘’π‘š||ξ€Έ||𝑀(𝑛)+π‘βˆ’π‘š||‖‖𝑀(𝑛)β‰₯βˆ’βˆ’π‘šβ€–β€–βˆ’πœ–πœ†1β€–β€–π‘’π‘šβ€–β€–β€–β€–π‘€βˆ’π‘šβ€–β€–βˆ’π‘βˆšπ‘‡βˆšπœ†1β€–β€–π‘€βˆ’π‘šβ€–β€–βˆΆ=βˆ’π‘5β€–β€–π‘€βˆ’π‘šβ€–β€–βˆ’π‘6β€–β€–π‘’π‘šβ€–β€–β€–β€–π‘€βˆ’π‘šβ€–β€–,(3.8) where 𝑐4=πœ†π‘˜πœ†π‘˜βˆ’1βˆ’1>0,𝑐5π‘βˆš=1+π‘‡βˆšπœ†1,𝑐6=πœ–πœ†1.(3.9) And hence we also have β€–β€–π‘€βˆ’π‘šβ€–β€–β€–β€–π‘’π‘šβ€–β€–βŸΆ0asπ‘šβŸΆβˆž.(3.10) By (3.7) and (3.10), we have β€–β€–π‘€π‘šβ€–β€–β€–β€–π‘’π‘šβ€–β€–β€–β€–π‘£βŸΆ0,π‘šβ€–β€–β€–β€–π‘’π‘šβ€–β€–βŸΆ1asπ‘šβŸΆβˆž.(3.11) By (𝐺+), there exist 𝛿>0 and π‘€βˆˆβ„• such that 𝑇𝑛=1ξ€·πΊξ…žξ€·π‘’π‘šξ€Έ(𝑛),π‘£π‘šξ€Έ(𝑛)β‰₯𝛿,βˆ€π‘šβ‰₯𝑀.(3.12) This implies that βŸ¨π½ξ…žξ€·π‘’π‘šξ€Έ,π‘£π‘šβŸ©=βˆ’π‘‡ξ“π‘›=1ξ€·πΊξ…žξ€·π‘’π‘šξ€Έ(𝑛),π‘£π‘šξ€Έ(𝑛)β‰€βˆ’π›Ώ,βˆ€π‘šβ‰₯𝑀,(3.13) and hence β€–β€–π½ξ…žξ€·π‘’π‘šξ€Έβ€–β€–β€–β€–π‘’π‘šβ€–β€–β‰₯β€–β€–π½ξ…žξ€·π‘’π‘šξ€Έβ€–β€–β€–β€–π‘£π‘šβ€–β€–β‰₯||βŸ¨π½ξ…žξ€·π‘’π‘šξ€Έ,π‘£π‘šβŸ©||β‰₯𝛿,βˆ€π‘šβ‰₯𝑀,(3.14) which is a contradiction to (3.2). Thus {π‘’π‘š} is bounded. The proof is complete.

Lemma 3.2. Let (1.8) and (1.9) hold. Then (1)πΆπ‘ž(𝐽,∞)β‰…π›Ώπ‘ž,π‘˜πΉ provided that (𝐺+) holds; (2)πΆπ‘ž(𝐽,∞)β‰…π›Ώπ‘ž,π‘˜βˆ’1𝐹 provided that (πΊβˆ’) holds.

Proof. We only prove the case (1). Define a bilinear function π‘Ž(𝑒,𝑣)=πœ†π‘˜π‘‡ξ“π‘›=1(𝑒(𝑛),𝑣(𝑛)),βˆ€π‘’,π‘£βˆˆπΈ.(3.15) Then by (2.7) we have ||||β‰€πœ†π‘Ž(𝑒,𝑣)π‘˜πœ†1‖𝑒‖‖𝑣‖.(3.16) And hence there exists a unique continuous bounded linear operator πΎβˆΆπΈβ†’πΈ such that βŸ¨πΎπ‘’,π‘£βŸ©=πœ†π‘˜π‘‡ξ“π‘›=1(𝑒(𝑛),𝑣(𝑛)).(3.17) Since βŸ¨πΎπ‘’,π‘’βŸ©βˆˆβ„ for all π‘’βˆˆπΈ, we can conclude that 𝐾 is a self-adjoint operator and 1𝐽(𝑒)=2⟨(πΌβˆ’πΎ)𝑒,π‘’βŸ©βˆ’π‘‡ξ“π‘›=1𝐺(𝑒(𝑛)).(3.18) Then 𝐽 has the form (2.16) with 𝑄(𝑒)=βˆ’π‘‡ξ“π‘›=1𝐺(𝑒(𝑛)),(3.19) and (1.11) implies that (2.17) holds. Let 𝐴=πΌβˆ’πΎ. Then ker𝐴=π‘Š0=span{πœ™π‘˜}. Next we show that (𝐺+) implies that the angle condition (π΄πΆβˆ’βˆž) at infinity holds.
If not, then for any π‘šβˆˆβ„• and each π›Όπ‘š=1/π‘š, there exists π‘’π‘š=π‘£π‘š+π‘€π‘šβˆˆπ‘Š0βŠ•(π‘Š+βŠ•π‘Šβˆ’) with π‘£π‘šβˆˆπ‘Š0, π‘€π‘šβˆˆπ‘Š+βŠ•π‘Šβˆ’ such that β€–β€–π‘’π‘šβ€–β€–β€–β€–π‘€β‰₯π‘š,π‘šβ€–β€–β‰€1π‘šβ€–β€–π‘’π‘šβ€–β€–ξ€·π‘’,(3.20)βŸ¨π½β€²π‘šξ€Έ,π‘£π‘šβŸ©>0.(3.21) On the other hand, (3.20) implies β€–β€–π‘’π‘šβ€–β€–β€–β€–π‘£βŸΆβˆž,π‘šβ€–β€–β€–β€–π‘’π‘šβ€–β€–βŸΆ1asπ‘šβŸΆβˆž.(3.22) Thus, by (𝐺+) there exist 𝛿>0 and π‘€βˆˆβ„• such that 𝑇𝑛=1ξ€·πΊξ…žξ€·π‘’π‘šξ€Έ(𝑛),π‘£π‘šξ€Έ(𝑛)β‰₯𝛿,βˆ€π‘šβ‰₯𝑀.(3.23) Therefore, βŸ¨π½ξ…žξ€·π‘’π‘šξ€Έ,π‘£π‘šβŸ©=βˆ’π‘‡ξ“π‘›=1ξ€·πΊξ…žξ€·π‘’π‘šξ€Έ(𝑛),π‘£π‘šξ€Έ(𝑛)β‰€βˆ’π›Ώ,βˆ€π‘šβ‰₯𝑀,(3.24) which is a contradiction to (3.21). Consequently (π΄πΆβˆ’βˆž) holds and by Lemma 3.1 and Proposition 2.5, πΆπ‘ž(𝐽,∞)=π›Ώπ‘ž,π‘˜πΉ. Similarly, we can prove that (2) holds.

In order to obtain a mountain pass point, we need the following lemmas.

Lemma 3.3. Let π‘‰ξ…ž+𝑉(𝑑)=ξ…žπ‘‰(𝑑),𝑑β‰₯0,0,𝑑≀0,ξ…žβˆ’ξƒ―π‘‰(𝑑)=ξ…ž(𝑑),𝑑≀0,0,𝑑β‰₯0,(3.25) and π‘‰Β±βˆ«(𝑑)=𝑑0𝑉′±(𝑠)𝑑𝑠. If lim|𝑑|β†’βˆžπ‘‰ξ…ž(𝑑)𝑑=π›Όβ‰ πœ†1,(3.26) then the functional 𝐽±1(𝑒)=2𝑇𝑛=0||||Δ𝑒(𝑛)2βˆ’π‘‡ξ“π‘›=1𝑉±(𝑒(𝑛))(3.27) satisfies the (PS) condition.

Proof. We only prove the case (𝐽+). Let {π‘’π‘š}βŠ‚πΈ such that π½ξ€·π‘’π‘šξ€ΈβŸΆπ‘βˆˆβ„,𝐽′+ξ€·π‘’π‘šξ€ΈβŸΆ0(3.28) as π‘šβ†’βˆž. Since 𝐸 is a finite-dimensional space, it suffices to show that {π‘’π‘š} is bounded in 𝐸. Suppose that {π‘’π‘š} is unbounded. Passing to a subsequence we may assume that β€–π‘’π‘šβ€–β†’βˆž and for each 𝑛, either |π‘’π‘š(𝑛)|β†’βˆž or {π‘’π‘š(𝑛)} is bounded.
Noticing that for all πœ‘βˆˆπΈ, βŸ¨π½β€²+ξ€·π‘’π‘šξ€Έ,πœ‘βŸ©=βŸ¨π‘’π‘š,πœ‘βŸ©βˆ’π‘‡ξ“π‘›=1𝑉′+ξ€·π‘’π‘šξ€Έξ€Έ.(𝑛),πœ‘(𝑛)(3.29)
Denote π‘€π‘šβˆΆ=π‘’π‘š/β€–π‘’π‘šβ€–, for a subsequence, π‘€π‘š converges to some 𝑀 with ‖𝑀‖=1. By (3.29), we have βŸ¨π½β€²+ξ€·π‘’π‘šξ€Έ,πœ‘βŸ©β€–β€–π‘’π‘šβ€–β€–=βŸ¨π‘€π‘š,πœ‘βŸ©βˆ’π‘‡ξ“π‘›=1𝑉′+ξ€·π‘’π‘šξ€Έ(𝑛)β€–β€–π‘’π‘šβ€–β€–ξƒͺ.,πœ‘(𝑛)(3.30) If |π‘’π‘š(𝑛)|β†’βˆž, then limπ‘šβ†’βˆžπ‘‰β€²+ξ€·π‘’π‘šξ€Έ(𝑛)π‘’π‘šπ‘€(𝑛)π‘š(𝑛)=𝛼𝑀+(𝑛),(3.31) where 𝑀+(𝑛)=max{𝑀(𝑛),0} with π‘›βˆˆβ„€(1,𝑇). If {π‘’π‘š(𝑛)} is bounded, then limπ‘šβ†’βˆžπ‘‰β€²+ξ€·π‘’π‘šξ€Έ(𝑛)β€–β€–π‘’π‘šβ€–β€–=0,𝑀(𝑛)=0.(3.32) Since 𝑀≠0, there is an 𝑛 for which |π‘’π‘š(𝑛)|β†’βˆž. So passing to the limit in (3.30), we have 𝑇𝑛=0(Δ𝑀(𝑛),Ξ”πœ‘(𝑛))βˆ’π›Όπ‘‡ξ“π‘›=1𝑀+ξ€Έ(𝑛),πœ‘(𝑛)=0.(3.33) This implies that 𝑀≠0 satisfies Ξ”2𝑀(π‘›βˆ’1)+𝛼𝑀+(𝑛)=0,π‘›βˆˆβ„€(1,𝑇),𝑀(0)=0=𝑀(𝑇+1).(3.34)
Now, we claim that 𝑀(𝑛)>0,βˆ€π‘›βˆˆβ„€(1,𝑇).(3.35)
In fact, let 𝑀𝑛0ξ€Έ=min{𝑀(𝑛)βˆΆπ‘›βˆˆβ„€(1,𝑇)}.(3.36) We only need to prove 𝑀(𝑛0)>0. If not, assume that 𝑀(𝑛0)≀0. Then by (3.34), we have Ξ”2𝑀(𝑛0βˆ’1)=0 and hence 𝑀(𝑛0βˆ’1)=𝑀(𝑛0)=𝑀(𝑛0+1). By induction, it is easy to get 𝑀(𝑛)=0 for all π‘›βˆˆβ„€(1,𝑇) which is a contradiction to 𝑀≠0 and hence (3.35) holds.
On the other hand, by Proposition 1.1 and Remark 1.2, we see that only the eigenfunction corresponding to the eigenvalue πœ†1 is positive, which is a contradiction to π›Όβ‰ πœ†1. The proof is complete.

Lemma 3.4. Under the conditions of Theorem 1.3, the functional 𝐽+ has a critical point 𝑒+>0 and πΆπ‘ž(𝐽+,𝑒+)β‰…π›Ώπ‘ž,1𝐹; the functional π½βˆ’ has a critical point π‘’βˆ’<0 and πΆπ‘ž(π½βˆ’,π‘’βˆ’)β‰…π›Ώπ‘ž,1𝐹.

Proof. We only prove the case of 𝐽+. Firstly, we prove that 𝐽+ satisfies the Mountain Pass Lemma and hence 𝐽+ has a nonzero critical point 𝑒+. In fact, 𝐽+∈𝐢1(𝐸,ℝ) and by Lemma 3.3 we see that 𝐽+ satisfies the (PS) condition. Clearly 𝐽+(0)=0. Thus we need to show that 𝐽+ satisfies (𝐽1) and (𝐽2). To verify (𝐽1), by (1.8) and (𝑉2), there exist 𝜌1>0 and 𝜌2>0 with π‘‰ξ…žξ…ž(0)<𝜌2<πœ†1 such that 1𝑉(𝑑)≀2𝜌2𝑑2(3.37) for |𝑑|β‰€πœŒ1. So, for all π‘’βˆˆπΈ, if βˆšβ€–π‘’β€–β‰€πœ†1𝜌1, then for each π‘›βˆˆβ„€(1,𝑇), |𝑒(𝑛)|β‰€πœŒ1 and 𝐽+1(𝑒)=2‖𝑒‖2βˆ’π‘‡ξ“π‘›=1𝑉+=1(𝑒(𝑛))2‖𝑒‖2βˆ’ξ“π‘›βˆˆπ‘1β‰₯1𝑉(𝑒(𝑛))2‖𝑒‖2βˆ’12𝜌2ξ“π‘›βˆˆπ‘1β‰₯1(𝑒(𝑛),𝑒(𝑛))2‖𝑒‖2βˆ’12𝜌2𝑇𝑛=1β‰₯1(𝑒(𝑛),𝑒(𝑛))2‖𝑒‖2βˆ’12𝜌2πœ†1‖𝑒‖2,(3.38) where 𝑁1={π‘›βˆˆβ„€(1,𝑇)βˆ£π‘’(𝑛)β‰₯0}. Let √𝜌=πœ†1𝜌11,π‘Ž=2ξ‚΅πœŒ1βˆ’2πœ†1ξ‚ΆπœŒ2.(3.39) Then 𝐽+(𝑒)|πœ•π΅πœŒβ‰₯π‘Ž>0 and hence (𝐽1) holds. For (𝐽2), by π‘‰ξ…žξ…ž(∞)=πœ†π‘˜βˆˆ(πœ†π‘˜βˆ’1,πœ†π‘˜+1), we claim that there exist 𝛾>πœ†π‘˜βˆ’1(β‰₯πœ†1), π‘βˆˆβ„ such that 𝑉(𝑑)β‰₯𝛾2𝑑2+𝑏,βˆ€π‘‘βˆˆβ„.(3.40)
In fact, by assumption (1.9), there exist 𝑀>0 and 𝑏1βˆˆβ„ such that 𝑉(𝑑)β‰₯(𝛾/2)𝑑2+𝑏1 for |𝑑|β‰₯𝑀. Meanwhile, there exists 𝑏2βˆˆβ„ such that 𝑉(𝑑)βˆ’(𝛾/2)𝑑2β‰₯𝑏2 for |𝑑|≀𝑀 by virtue of the continuity of 𝑉. Let 𝑏=min{𝑏1,𝑏2}, we get the conclusion.
Thus, if we choose π‘’βˆˆspan{πœ™1} with 𝑒>0 and ‖𝑒‖=1, then 𝐽+𝑑(𝑑𝑒)=22βˆ’π‘‡ξ“π‘›=1≀𝑑𝑉(𝑑𝑒(𝑛))22βˆ’π›Ύπ‘‘22=𝑑(𝑒,𝑒)βˆ’π‘π‘‡22βˆ’π›Ύπ‘‘22πœ†1βˆ’π‘π‘‡βŸΆβˆ’βˆž(3.41) as 0<𝑑→+∞. Thus, we can choose a constant 𝑑 large enough with 𝑑>𝜌 and 𝑒0=π‘‘π‘’βˆˆπΈ such that 𝐽+(𝑒0)≀0. (𝐽2) holds.
Therefore, by Theorem 2.7, 𝐽+ has a critical point 𝑒+β‰ 0 and similar to the proof of Lemma 3.3, we can prove that 𝑒+>0. So 𝑒+ is also a critical point of 𝐽. In the following we compute the critical group πΆπ‘ž(𝐽+,𝑒+) by using Theorem 2.9.
Assume that βŸ¨π½ξ…žξ…žξ€·π‘’+𝑣,π‘£βŸ©=βŸ¨π‘£,π‘£βŸ©βˆ’π‘‡ξ“π‘›=1ξ€·π‘‰ξ…žξ…žξ€·π‘’+ξ€Έξ€Έ(𝑛)𝑣(𝑛),𝑣(𝑛)β‰₯0,βˆ€π‘£βˆˆπΈ,(3.42) and that there exists 𝑣0β‰’0 such that βŸ¨π½ξ…žξ…žξ€·π‘’+𝑣0,π‘£βŸ©=0,βˆ€π‘£βˆˆπΈ.(3.43) This implies that 𝑣0 satisfies Ξ”2𝑣0(π‘›βˆ’1)+π‘‰ξ…žξ…žξ€·π‘’+(𝑣𝑛)0(𝑣𝑛)=0,π‘›βˆˆβ„€(1,𝑇),0(0)=𝑣0(𝑇+1)=0.(3.44) Hence the eigenvalue problem Ξ”2𝑣(π‘›βˆ’1)+πœ†π‘‰ξ…žξ…žξ€·π‘’+(𝑛)𝑣(𝑛)=0,π‘›βˆˆβ„€(1,𝑇),𝑣(0)=𝑣(𝑇+1)=0(3.45) has an eigenvalue πœ†=1. (𝑉1) implies that 1 must be a simple eigenvalue; see [4]. So, dimker(π½ξ…žξ…ž(𝑒0))=1. Since 𝐸 is a finite-dimensional Hilbert space, the Morse index of 𝑒+ must be finite and π½ξ…žξ…ž(𝑒+) must be a Fredholm operator. By Theorem 2.9, πΆπ‘ž(𝐽+,𝑒+)β‰…π›Ώπ‘ž,1𝐹. The proof is complete.

Remark 3.5. We can choose the neighborhood π‘ˆ of 𝑒+ such that 𝑒>0 for all π‘’βˆˆπ‘ˆ. Therefore, πΆπ‘žξ€·π½,𝑒+ξ€Έβ‰…πΆπ‘žξ€·π½+,𝑒+ξ€Έβ‰…π›Ώπ‘ž,1𝐹.(3.46) Similarly, πΆπ‘ž(𝐽,π‘’βˆ’)β‰…πΆπ‘ž(π½βˆ’,π‘’βˆ’)β‰…π›Ώπ‘ž,1𝐹.(3.47)

Now, we give the proof of Theorem 1.3.

Proof of Theorem 1.3. We only prove the case (i). By Lemma 3.2, πΆπ‘ž(𝐽,∞)β‰…π›Ώπ‘ž,π‘˜πΉ.(3.48) Hence by Proposition 2.4 the functional 𝐽 has a critical point 𝑒1 satisfying πΆπ‘˜ξ€·π½,𝑒1≇0.(3.49) Since βŸ¨π½ξ…žξ…žξ‚΅π‘‰(0)𝑒,π‘’βŸ©β‰₯1βˆ’ξ…žξ…ž(0)πœ†1‖𝑒‖2,(3.50) by (𝑉2) and 𝐽(0)=π½ξ…ž(0)=0, we see that 0 is a local minimum of 𝐽. Hence πΆπ‘ž(𝐽,0)β‰…π›Ώπ‘ž,0𝐹.(3.51) By Remark 3.5, (3.49), (3.51), and π‘˜β‰₯2 we get that 𝑒+, π‘’βˆ’, and 𝑒1 are three nonzero critical points of 𝐽 with 𝑒+>0 and π‘’βˆ’<0. The proof is complete.

4. An Example and Future Directions

To illustrate the use of Theorem 1.3, we offer the following example.

Example 4.1. Consider the BVP Ξ”2𝑒(π‘›βˆ’1)+π‘‰ξ…ž(𝑒(𝑛))=0,π‘›βˆˆβ„€(1,5),𝑒(0)=0=𝑒(6),(4.1) where π‘‰βˆˆπΆ2(ℝ,ℝ) is defined as follows: ⎧βŽͺ⎨βŽͺ⎩1𝑉(𝑑)=𝑑1021,|𝑑|≀1,2𝑑2+34𝑑4/3,|𝑑|β‰₯10,astrictlyconvexfunction,otherwise.(4.2) It is easy to verify that 𝑉 satisfies (1.8), (1.9), (1.11), (𝑉1), and (𝑉2) with π‘˜=2. To verify the condition (𝐺+), note that πΊξ…ž(𝑑)=𝑑1/3 for |𝑑|β‰₯10, we claim that 5𝑛=1ξ€·πΊξ…žξ€·π‘’π‘šξ€Έ(𝑛),π‘£π‘šξ€Έ(𝑛)⟢+∞,asπ‘šβŸΆβˆž(4.3) which implies that (𝐺+) holds.
To this end, for any constant π‘Ÿ>1, we introduce another norm in 𝐸(𝑇=5) as follows: β€–π‘’β€–π‘Ÿ=5𝑛=1||||𝑒(𝑛)π‘Ÿξƒͺ1/π‘Ÿ,βˆ€π‘’βˆˆπΈ.(4.4) Since 𝐸 is finite dimensional, there exist two constants 𝐢2β‰₯𝐢1>0 such that 𝐢1β€–π‘’β€–β‰€β€–π‘’β€–π‘Ÿβ‰€πΆ2‖𝑒‖,βˆ€π‘’βˆˆπΈ.(4.5)
Now, by (𝐺+), for any πœ– small enough, it is easy to see that β€–β€–π‘€π‘šβ€–β€–β€–β€–π‘’β‰€πœ–π‘šβ€–β€–(4.6) holds for π‘š large enough.
Set Ξ©1=ξ€½||π‘’π‘›βˆˆβ„€(1,5)βˆΆπ‘š||ξ€Ύ(𝑛)β‰₯10,Ξ©2=β„€(1,5)⧡Ω1.(4.7) Since β€–π‘’π‘šβ€–β†’βˆž, Ξ©1β‰ βˆ…, for π‘š large enough. And for π‘š large enough, we have 5𝑛=1ξ€·πΊξ…žξ€·π‘’π‘šξ€Έ(𝑛),π‘£π‘šξ€Έ=(𝑛)π‘›βˆˆΞ©1ξ€·π‘’π‘š1/3(𝑛),π‘£π‘šξ€Έ+(𝑛)π‘›βˆˆΞ©2ξ€·πΊξ…žξ€·π‘’π‘šξ€Έ(𝑛),π‘’π‘šξ€Έβˆ’ξ“(𝑛)π‘›βˆˆΞ©2ξ€·ξ€·π‘’πΊβ€²π‘šξ€Έ(𝑛),π‘€π‘šξ€Έβ‰₯(𝑛)π‘›βˆˆΞ©1ξ€·π‘’π‘š1/3(𝑛),π‘£π‘š(‖‖𝑒𝑛)βˆ’π‘βˆ’π‘πœ–π‘šβ€–β€–=ξ“π‘›βˆˆΞ©1ξ€·π‘’π‘š1/3(𝑛),π‘’π‘šξ€Έβˆ’ξ“(𝑛)π‘›βˆˆΞ©1ξ€·π‘’π‘š1/3(𝑛),π‘€π‘šξ€Έβ€–β€–π‘’(𝑛)βˆ’π‘βˆ’π‘πœ–π‘šβ€–β€–.(4.8) Here and below we denote by 𝑐 various positive constants. Since ξ“π‘›βˆˆΞ©1ξ€·π‘’π‘š1/3(𝑛),π‘’π‘šξ€Έ=(𝑛)5𝑛=1ξ€·π‘’π‘š1/3(𝑛),π‘’π‘šξ€Έβˆ’ξ“(𝑛)π‘›βˆˆΞ©2ξ€·π‘’π‘š1/3(𝑛),π‘’π‘šξ€Έ=‖‖𝑒(𝑛)π‘šβ€–β€–4/34/3ξ“βˆ’π‘,π‘›βˆˆΞ©1ξ€·π‘’π‘š1/3(𝑛),π‘€π‘šξ€Έβ‰€(𝑛)5𝑛=1||π‘’π‘š||(𝑛)1/3||π‘€π‘š||≀(𝑛)5𝑛=1||π‘’π‘š||(𝑛)4/3ξƒͺ1/45𝑛=1||π‘€π‘š||(𝑛)4/3ξƒͺ3/4=β€–β€–π‘’π‘šβ€–β€–1/34/3β€–β€–π‘€π‘šβ€–β€–4/3β€–β€–π‘’β‰€π‘πœ–π‘šβ€–β€–4/34/3.(4.9) Hence 5𝑛=1ξ€·πΊξ…žξ€·π‘’π‘šξ€Έ(𝑛),π‘£π‘šξ€Έβ€–β€–π‘’(𝑛)β‰₯(1βˆ’π‘πœ–)π‘šβ€–β€–4/34/3β€–β€–π‘’βˆ’π‘πœ–π‘šβ€–β€–βˆ’π‘.(4.10) Since πœ– is small enough, we get (4.3) holds by the above and (4.5). Hence, by Theorem 1.3, BVP (4.1) has at least three nontrivial solutions.

Morse theory has been proved very useful in proving the existence and multiplicity of solutions of operator equations with variational frameworks. However, it is well known that the minimax methods is also a useful tool for the same purpose. The advantage of the minimax methods is that it provides an estimate of the critical value. But it is hard to distinguish critical points obtained by this methods with those by other methods, if the local behavior of the critical points is not very well known. However, critical groups serve as a topological tool in distinguishing isolated critical points. Hence, in order to obtain multiple solutions by using Morse theory, it is crucial to describe critical groups clearly.

A natural question is: can we use the same methods in this paper to other BVPs? Noticing that the key conditions which guarantee the multiplicity of solutions of the BVP (1.1) are as follows:

(1)the BVP has a variational framework;(2)the eigenvalues of the corresponding linear BVP are nonzero and there is a one-sign eigenfunction,

hence, if the difference equation

Ξ”2𝑒(π‘›βˆ’1)+𝑉′(𝑒(𝑛))=0,π‘›βˆˆβ„€(1,𝑇)(4.11) subject to some other boundary value conditions satisfying (1) and (2), then we can obtain similar results to Theorem 1.3.

Example 4.2. Consider the BVP Ξ”2𝑒(π‘›βˆ’1)+π‘‰ξ…ž(𝑒𝑒(𝑛))=0,π‘›βˆˆβ„€(1,𝑇),(0)=0=Δ𝑒(𝑇).(4.12) Let 𝐸=π‘’βˆΆπ‘’={𝑒(𝑛)}𝑇+1𝑛=0ξ€Ύ.with𝑒(0)=0=Δ𝑒(𝑇)(4.13) Then 𝐸 is a 𝑇-dimensional Hilbert space with the inner product βŸ¨π‘’,π‘£βŸ©=𝑇𝑛=0(Δ𝑒(𝑛),Δ𝑣(𝑛)).(4.14) Define the functional 𝐽 on 𝐸 by 𝐽(𝑒)=𝑇𝑛=012||||Δ𝑒(𝑛)2βˆ’π‘‡ξ“π‘›=1𝑉(𝑒(𝑛)).(4.15) It is easy to see that 𝑒 is a critical point of 𝐽 in 𝐸 if and only if 𝑒 is a solution of the BVP (4.12).
The eigenvalues of the linear BVP Ξ”2𝑒(π‘›βˆ’1)+πœ†π‘’(𝑛)=0,π‘›βˆˆβ„€(1,𝑇),𝑒(0)=0=Δ𝑒(𝑇)(4.16) are πœ†=πœ†π‘™=4sin2π‘™πœ‹2(2𝑇+1),𝑙=1,2,…,𝑇,(4.17) and the corresponding eigenfunctions are πœ™π‘™(𝑛)=sinπ‘™πœ‹π‘›2𝑇+1,𝑙=1,2,…,𝑇.(4.18) Hence, πœ†π‘™β‰ 0 for all π‘™βˆˆβ„€(1,𝑇) and πœ™1(𝑛)>0 for all π‘›βˆˆβ„€(1,𝑇). Therefore, the BVP (4.12) satisfies (1) and (2) and hence we can obtain similar results as in Theorem 1.3.

However, consider the following difference BVP:

Ξ”2𝑒(π‘›βˆ’1)+𝑉′(𝑒(𝑛))=0,π‘›βˆˆβ„€(1,𝑇),𝑒(0)=𝑒(𝑇),Δ𝑒(0)=Δ𝑒(𝑇).(4.19) It is easy to verify that the variational functional of the BVP (4.19) is

𝐽(𝑒)=𝑇𝑛=110π‘₯0200𝑑2||||Δ𝑒(𝑛)2ξ‚„βˆ’π‘‰(𝑒(𝑛)),βˆ€π‘’βˆˆπΈ1,(4.20) where

𝐸1=ξ€½π‘’βˆΆπ‘’={𝑒(𝑛)}𝑇+1𝑛=0ξ€Ύ.with𝑒(0)=𝑒(𝑇),Δ𝑒(0)=Δ𝑒(𝑇+1)(4.21) But, πœ†=0 is an eigenvalue of the linear BVP:

Ξ”2𝑒(π‘›βˆ’1)+πœ†π‘’(𝑛)=0,π‘›βˆˆβ„€(1,𝑇),𝑒(0)=𝑒(𝑇),Δ𝑒(0)=Δ𝑒(𝑇).(4.22) So, for the BVP (4.19), we need to find other techniques (e.g., dual variational methods if possible) to study the BVP (4.19).

Acknowledgments

The authors would like to express their thanks to the referees for helpful suggestions. This research is supported by Guangdong College Yumiao Project (2009).

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Copyright © 2010 Bo Zheng and Huafeng Xiao. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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