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

Abstract

This paper studies the existence of multiple solutions of the second-order difference boundary value problem Δ²𝑢(𝑛−1)+𝑉^(𝑢(𝑛))=0, 𝑛∈ℤ(1,𝑇), 𝑢(0)=0=𝑢(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 𝜆_𝑘(𝑘≥2) of linear difference problem Δ²𝑢(𝑛−1)+𝜆𝑢(𝑛)=0, 𝑛∈ℤ(1,𝑇), 𝑢(0)=0=𝑢(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, ℕ, and ℤ be the sets of real numbers, natural numbers, and integers, respectively. For any 𝑎,𝑏∈ℤ, 𝑎≤𝑏, define ℤ(𝑎,𝑏)={𝑎,𝑎+1,…,𝑏}.

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

Δ²𝑢(𝑛−1)𝑢+(0𝑉)^=(𝑢0(𝑛=))𝑢=(𝑇0+,1𝑛),∈ℤ(1,𝑇),(1.1) where 𝑉∈𝐶²(ℝ,ℝ) and Δ denotes the forward difference operator defined by Δ𝑢(𝑛)=𝑢(𝑛+1)−𝑢(𝑛), Δ²𝑢(𝑛)=Δ(Δ𝑢(𝑛)).

By a solution 𝑢 of the BVP (1.1), we mean a real sequence {𝑢(𝑛)}^𝑇+1𝑛=0(=(𝑢(0),𝑢(1),…,𝑢(𝑇+1))) satisfying the BVP (1.1). For 𝑢={𝑢(𝑛)}^𝑇+1𝑛=0 with 𝑢(0)=0=𝑢(𝑇+1), we say that 𝑢≠0 if there exists at least one 𝑛∈ℤ(1,𝑇) such that 𝑢(𝑛)≠0. We say that 𝑢 is positive (and write 𝑢>0) if for all 𝑛∈ℤ(1,𝑇), 𝑢(𝑛)>0, and similarly, 𝑢 is negative (𝑢<0) if for all 𝑛∈ℤ(1,𝑇), 𝑢(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:

Δ²𝑢(𝑛−1)+𝑢(𝑓0)(𝑛=,𝑢0(𝑛=))𝑢=(𝑇0+,1𝑛)∈ℤ(1,𝑇),(1.2) and obtained the existence of multiple positive solutions, where 𝑓 may be singular at 𝑢=0. In [2], Jiang and Zhou employed the Mountain Pass Lemma together with strongly monotone operator principle, to study the following difference BVP:

Δ²𝑢(𝑛−1)+𝑢𝑓(0()𝑛,=𝑢0(𝑛=))Δ=𝑢0(,𝑇)𝑛∈ℤ(1,𝑇),(1.3) and obtained existence and uniqueness results, where 𝑓∶ℤ(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)(Δ𝑢(𝑛−1)Δ)^𝛿𝑢(+0)𝑞=(𝑛𝐴)𝑢,^𝛿(𝑢𝑛)(𝑇=+𝑓1()𝑛=,𝑢𝐵(𝑛)),𝑛∈ℤ(1,𝑇),(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 {𝑞(𝑛)}^𝑇𝑛=1 are real sequences, 𝑝(𝑛)≠0 for all 𝑛∈ℤ(1,𝑇+1), and 𝐴, 𝐵 are two given constants, 𝑓∶ℤ(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:

Δ²𝑢(𝑛−1𝑢)(+0)𝜆=𝑢(0𝑛=)=𝑢(0𝑇,+𝑛1∈).ℤ(1,𝑇),(1.5)

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

Proposition 1.1. The eigenvalues of (1.5) are 𝜆=𝜆_𝑙=4sin²𝑙𝜋2(𝑇+1),𝑙=1,2,…,𝑇,(1.6) and the corresponding eigenfunction with 𝜆_𝑙 is 𝜙_𝑙(𝑛)=sin(𝑙𝜋𝑛/(𝑇+1)), 𝑙=1,2,…,𝑇.

Remark 1.2. (1) The set of functions {𝜙_𝑙(𝑛),𝑙=1,2,…,𝑇} is orthogonal on ℤ(1,𝑇) with respect to the weight function 𝑟(𝑛)≡1, that is, ^𝑇_𝑛=1𝜙_𝑙(𝑛),𝜙_𝑗(𝑛)=0∀𝑙≠𝑗.(1.7) Moreover, for each 𝑙∈ℤ(1,𝑇), ∑^𝑇𝑛=1sin²(𝑙𝜋𝑛/(𝑇+1))=(𝑇+1)/2.
(2) It is easy to see that 𝜙₁ is positive and 𝜙_𝑙 changes sign for each 𝑙∈ℤ(2,𝑇), that is, {𝑛∶𝜙_𝑙(𝑛)>0}≠∅ and {𝑛∶𝜙_𝑙(𝑛)<0}≠∅.

For (1.1), we assume that

𝑉𝑉(0)=𝑉^(0)=0,(1.8)_(∞)∶=lim_|𝑡|→∞𝑉^(𝑡)𝑡=𝜆^𝑘,(1.9) where 𝜆_𝑘 is an eigenvalue of (1.5). Hence the BVP (1.1) has a trivial solution 𝑢≡0. And we say that BVP (1.1) is resonant at infinity if (1.9) holds.

Let

𝑊⁻=span𝜙₁,𝜙₂,…,𝜙_𝑘−1,𝑊⁰=span𝜙_𝑘,𝑊⁺=span𝜙_𝑘+1,𝜙_𝑘+2,…,𝜙_𝑇.(1.10) Let 𝐺(𝑡)=∫^𝑡0𝐺^(𝑠)𝑑𝑠=𝑉(𝑡)−(𝜆_𝑘/2)𝑡². By (1.9) we have

lim_|𝑡|→∞𝐺^(𝑡)𝑡=0.(1.11) Assume that the following conditions on 𝐺^(𝑡) hold.

(𝐺^±) If ‖𝑢_𝑚‖→∞ such that ‖𝑣_𝑚‖/‖𝑢_𝑚‖→1, then there exist 𝛿>0 and 𝑀∈ℕ such that

±^𝑇_𝑛=1𝐺^𝑢_𝑚(𝑛),𝑣_𝑚(𝑛)≥𝛿,∀𝑚≥𝑀,(1.12) where 𝑢_𝑚=𝑣_𝑚+𝑤_𝑚, 𝑣_𝑚∈𝑊⁰, 𝑤_𝑚∈𝑊∶=𝑊⁺⊕𝑊⁻.

The main result of this paper is as follows.

Theorem 1.3. Let (1.8), (1.9) hold and (𝑉1)𝑉^(𝑡)>0 for all 𝑡∈ℝ, (𝑉2)𝑉^(0)<𝜆₁hold. 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 𝑘≥2; (ii)(G⁻) and 𝑘≥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

lim_{‖𝑣‖→∞}||𝐺^(𝑧)||≤𝑐₁|𝑧|^𝑠+𝑐₂,(1.13)inf_𝑣∈𝑊01‖𝑣‖^2𝑠𝐺(𝑧)≥4𝛽²𝛿𝑇,(1.14) where 𝑐₁>0, 𝑐₂>0, 𝑠∈(0,1),𝛽=𝑐₁𝑇^(1−𝑠)/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=𝑢∶𝑢={𝑢(𝑛)}^𝑇+1𝑛=0with𝑢(0)=0=𝑢(𝑇+1)∈ℝ.(2.1)𝐸 can be equipped with the norm ‖⋅‖ and the inner product ⟨⋅,⋅⟩ as follows:

‖𝑢‖=^𝑇_𝑛=0⟨𝑢,𝑣⟩=_𝑇||Δ𝑢(𝑛)||²0𝑥0200𝑑^1/2,∀𝑢∈𝐸,_𝑛=0(Δ𝑢(𝑛),Δ𝑣(𝑛)),∀𝑢,𝑣∈𝐸,(2.2) where |⋅| denotes the Euclidean norm in ℝ and (⋅,⋅) denotes the usual scalar product in ℝ. It is easy to see that (𝐸,⟨⋅,⋅⟩) is a Hilbert space. Consider the functional defined on 𝐸 by

𝐽(𝑢)=12^𝑇_𝑛=0|Δ𝑢(𝑛)|²−^𝑇_𝑛=1𝑉(𝑢(𝑛)).(2.3) We claim that if 𝑢∈𝐸 is a critical point of 𝐽, then 𝑢 is precisely a solution of the BVP (1.1). Indeed, for every 𝑢,𝑣∈𝐸, we have

⟨𝐽^(𝑢),𝑣⟩=^𝑇_𝑛=0(Δ𝑢(𝑛),Δ𝑣(𝑛))−^𝑇_𝑛=1𝑉^(𝑢(𝑛)),𝑣(𝑛)=−^𝑇_𝑛=1Δ²𝑢(𝑛−1)+𝑉^(𝑢(𝑛)),𝑣(𝑛).(2.4)

So, if 𝐽^(𝑢)=0, then we have

^𝑇_𝑛=1Δ²𝑢(𝑛−1)+𝑉^(𝑢(𝑛)),𝑣(𝑛)=0.(2.5) Since 𝑣∈𝐸 is arbitrary, we obtain

Δ²𝑢(𝑛−1)+𝑉^(𝑢(𝑛))=0,𝑛∈ℤ(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=𝑊⁻⊕𝑊⁰⊕𝑊⁺. For all 𝑢∈𝐸, denote 𝑢=𝑤⁰+𝑤⁺+𝑤⁻ with 𝑤⁰∈𝑊⁰, 𝑤⁺∈𝑊⁺, and 𝑤⁻∈𝑊⁻, then we have the following Wirtinger type inequalities:

𝜆_1𝑇_𝑛=1(𝑢(𝑛),𝑢(𝑛))≤‖𝑢‖²≤𝜆_𝑇𝑇_𝑛=1𝜆(𝑢(𝑛),𝑢(𝑛)),∀𝑢∈𝐸,(2.7)_1𝑇_𝑛=1(𝑤⁻(𝑛),𝑤⁻(𝑛))≤‖𝑤⁻‖²≤𝜆_𝑇𝑘−1_𝑛=1𝜆(𝑤⁻(𝑛),𝑤⁻(𝑛)),∀𝑤⁻∈𝑊⁻,(2.8)_𝑇𝑘+1_𝑛=1𝑤⁺(𝑛),𝑤⁺(𝑛)≤‖‖𝑤⁺‖‖²≤𝜆_𝑇𝑇_𝑛=1𝑤⁺(𝑛),𝑤⁺(𝑛),∀𝑤⁺∈𝑊⁺,(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∈𝐶¹(𝐸,ℝ). Denote

𝐽^𝑐={𝑢∈𝐸∶𝐽(𝑢)≤𝑐},𝒦_𝑐={𝑢∈𝐸∶𝐽′(𝑢)=0,𝐽(𝑢)=𝑐}(2.10) for 𝑐∈ℝ. 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 {𝑢_𝑚}⊂𝐸 such that {𝐽(𝑢_𝑚)} is bounded and 𝐽′(𝑢_𝑚)→0 as 𝑚→∞ 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 ((𝐶) condition) if any sequence {𝑢_𝑚}⊂𝐸 such that {𝐽(𝑢_𝑚)} is bounded and (1+‖𝑢_𝑚‖)‖𝐽^(𝑢_𝑚)‖→0 as 𝑚→∞ has a convergent subsequence.

If 𝐽 satisfies the (PS) condition or the (𝐶) 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 (𝐷_𝑐) condition at the level 𝑐∈ℝ if for any 𝜖>0 and any neighborhood 𝒩 of 𝒦_𝑐, there are 𝜖>0 and a continuous deformation 𝜂∶[0,1]×𝐸→𝐸 such that(1)𝜂(0,𝑢)=𝑢 for all 𝑢∈𝐸;(2)𝜂(𝑡,𝑢)=𝑢 for all 𝑢∉𝐽⁻¹([𝑐−𝜖,𝑐+𝜖]);(3)𝐽(𝜂(𝑡,𝑢)) is nonincreasing in 𝑡 for any 𝑢∈𝐸;(4)𝜂(1,𝐽^𝑐+𝜖⧵𝒩)⊂𝐽^𝑐−𝜖. 𝐽 satisfies the (𝐷) condition if 𝐽 satisfies the (𝐷_𝑐) condition for all 𝑐∈ℝ.

Let 𝑢₀ be an isolated critical point of 𝐽 with 𝐽(𝑢₀)=𝑐∈ℝ, and let 𝑈 be a neighborhood of 𝑢₀, the group

𝐶_𝑞𝐽,𝑢₀∶=𝐻_𝑞𝐽^𝑐∩𝑈,𝐽^𝑐∩𝑈⧵𝑢₀,𝑞∈ℤ,(2.11) is called the 𝑞th critical group of 𝐽 at 𝑢₀, where 𝐻_𝑞(𝐴,𝐵) denotes the 𝑞th singular relative homology group of the pair (𝐴,𝐵) over a field 𝐹, which is defined to be quotient 𝐻_𝑞(𝐴,𝐵)=𝑍_𝑞(𝐴,𝐵)/𝐵_𝑞(𝐴,𝐵), where 𝑍_𝑞(𝐴,𝐵) is the 𝑞th singular relative closed chain group and 𝐵_𝑞(𝐴,𝐵) is the 𝑞th singular relative boundary chain group.

Let 𝒦={𝑢∈𝐸∶𝐽′(𝑢)=0}. If 𝐽(𝒦) is bounded from below by 𝑎∈ℝ and 𝐽 satisfies the (𝐷_𝑐) condition for all 𝑐≤𝑎, then the group

𝐶_𝑞(𝐽,∞)∶=𝐻_𝑞(𝐸,𝐽^𝑎),𝑞∈ℤ,(2.12) is called the 𝑞th critical group of 𝐽 at infinity [11].

Assume that #𝒦<∞ and 𝐽 satisfies the (𝐷) condition. The Morse-type numbers of the pair (𝐸,𝐽^𝑎) are defined by

𝑀_𝑞=𝑀_𝑞(𝐸,𝐽^𝑎)=_𝑢∈𝒦dim𝐶_𝑞(𝐽,𝑢),(2.13) and the Betti numbers of the pair (𝐸,𝐽^𝑎) are

𝛽_𝑞∶=dim𝐶_𝑞(𝐽,∞).(2.14) By Morse theory [12, 13], the following relations hold:

^𝑞_𝑗=0(−1)^𝑞−𝑗𝑀_𝑗≥^𝑞_𝑗=0(−1)^𝑞−𝑗𝛽_𝑗,𝑞∈ℤ,_∞_𝑞=0𝑀_𝑞=^∞_𝑞=0𝛽_𝑞.(2.15)

Thus, if 𝐶_𝑞(𝐽,∞)≇0, for some 𝑘∈ℤ, then there must exist a critical point 𝑢 of 𝐽 with 𝐶_𝑞(𝐽,𝑢)≇0, which can be rephrased as follows.

Proposition 2.4. Let 𝐸 be a real Hilbert space and 𝐽∈𝐶²(𝐸,ℝ). Assume that #𝒦<∞ and that 𝐽 satisfies the (𝐷) condition. If there exists some 𝑞∈ℤ such that 𝐶_𝑞(𝐽,∞)≇0, then 𝐽 must have a critical point 𝑢 with 𝐶_𝑞(𝐽,𝑢)≇0.

In order to prove our main result, we need the following result about the critical group on 𝐶_𝑞(𝐽,∞).

Proposition 2.5. Let the functional 𝐽∶𝐸→ℝ be of the form 𝐽(𝑢)=12⟨𝐴𝑢,𝑢⟩+𝑄(𝑢),(2.16) where 𝐴∶𝐸→𝐸 is a self-adjoint linear operator such that 0 is isolated in 𝜎(𝐴), the spectrum of 𝐴. Assume that 𝑄∈𝐶¹(𝐸,ℝ) satisfies lim_{‖𝑢‖→∞}‖‖𝑄^(𝑢)‖‖‖𝑢‖=0.(2.17) Denote 𝑉∶=ker𝐴, 𝑊∶=𝑉^⟂=𝑊⁺⊕𝑊⁻, where 𝑊⁺ (𝑊⁻) is the subspace of 𝐸 on which 𝐴 is positive (negative) definite. Assume that 𝜇∶=dim𝑊⁻, 𝜈∶=dim𝑉≠0 are finite and that 𝐽 satisfies the (𝐷) condition. Then 𝐶_𝑞(𝐽,∞)≅𝛿_𝑞,𝑘^±𝐹,𝑞∈ℤ,(2.18) provided that 𝐽 satisfies the angle conditions at infinity. (𝐴𝐶^±∞): there exist 𝑀>0 and 𝛼∈(0,1) such that ±⟨𝐽′(𝑢),𝑣⟩≥0for𝑢=𝑣+𝑤,‖𝑢‖≥𝑀,‖𝑤‖≤𝛼‖𝑢‖,(2.19) where 𝑘⁺=𝜇, 𝑘⁻=𝜇+𝜈, 𝑣∈𝑉, and 𝑤∈𝑊.

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 𝐵_𝜌 denote the open ball in 𝐸 about 0 of radius 𝑟 and let 𝜕𝐵_𝜌 denote its boundary.

Theorem 2.7 (mountain pass lemma). Let 𝐸 be a real Banach space and 𝐽∈𝐶¹(𝐸,ℝ) satisfying the (PS) condition. Suppose 𝐽(0)=0 and that (𝐽1) there are constants 𝜌>0,𝑎>0 such that 𝐽|_𝜕𝐵𝜌≥𝑎>0, (𝐽2) there is a 𝑢₀∈𝐸⧵𝐵_𝜌 such that 𝐽(𝑢₀)≤0,then 𝐽 possesses a critical value 𝑐≥𝑎. Moreover 𝑐 can be characterized as 𝑐=inf_ℎ∈Γsup_𝑠∈[0,1]𝐽(ℎ(𝑠)),(2.20) where Γ=ℎ∈𝐶([0,1],𝐸)∣ℎ(0)=0,ℎ(1)=𝑢₀.(2.21)

Definition 2.8 (mountain pass point). An isolated critical point 𝑢 of 𝐽 is called a mountain pass point, if 𝐶₁(𝐽,𝑢)≇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∈𝐶²(𝐸,ℝ) has a mountain pass point 𝑢, and that 𝐽^(𝑢) is a Fredholm operator with finite Morse index, satisfying 𝐽^𝑢₀≥0,0∈𝜎𝐽^𝑢₀⟹dimker𝐽^𝑢₀=1,(2.22) then 𝐶_𝑞𝐽,𝑢₀≅𝛿_𝑞,1𝐹,𝑞∈ℤ.(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 (𝐶) condition (Lemma 3.1) and compute the critical group 𝐶_𝑞(𝐽,∞) (Lemma 3.2). Then, we employ the cut-off technique and the Mountain Pass Lemma to obtain two critical points 𝑢⁺,𝑢⁻ of 𝐽 and compute the critical groups 𝐶_𝑞(𝐽,𝑢⁺) and 𝐶_𝑞(𝐽,𝑢⁻) (Lemmas 3.3 and 3.4). Finally, we prove Theorem 1.3.

Rewrite the functional 𝐽 as

𝐽(𝑢)=12^𝑇_𝑛=0|Δ𝑢(𝑛)|²−𝜆^𝑘2^𝑇_𝑛=1|𝑢(𝑛)|²−^𝑇_𝑛=1𝐺(𝑢(𝑛)),∀𝑢∈𝐸.(3.1)

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

Proof. We only prove the case where (𝐺⁺) holds. Let {𝑢_𝑚}⊂𝐸 such that 𝐽𝑢_𝑚⟶𝑐∈ℝ,1+‖‖𝑢_𝑚‖‖‖‖𝐽^𝑢_𝑚‖‖⟶0as𝑚⟶∞.(3.2) Then for all 𝜑∈𝐸, we have ⟨𝐽^𝑢_𝑚,𝜑⟩=⟨𝑢_𝑚,𝜑⟩−𝜆_𝑘𝑇_𝑛=1𝑢_𝑚(𝑛),𝜑(𝑛)−^𝑇_𝑛=1𝐺^𝑢_𝑚(𝑛),𝜑(𝑛).(3.3) Denote 𝑢_𝑚=𝑣_𝑚+𝑤^+𝑚+𝑤^−𝑚 with 𝑣_𝑚∈𝑊⁰, 𝑤^+𝑚∈𝑊⁺ and 𝑤^−𝑚∈𝑊⁻. Since 𝐸 is a finite-dimensional Hilbert space, it suffices to show that {𝑢_𝑚} is bounded. Suppose that {𝑢_𝑚} is unbounded. Passing to a subsequence we may assume that ‖𝑢_𝑚‖→∞ as 𝑚→∞.
By (1.11), for any 𝜖>0, there exists 𝑏∈ℝ such that ||𝐺^(𝑡)||≤𝜖|𝑡|+𝑏,∀𝑡∈ℝ.(3.4) Let 𝜑=𝑤^+𝑚 in (3.3). Then by (2.7), (2.9), and (3.4), we have 𝑐₁‖‖𝑤^+𝑚‖‖²∶=1−𝜆^𝑘𝜆_𝑘+1‖‖𝑤^+𝑚‖‖²≤‖‖𝑤^+𝑚‖‖²−𝜆_𝑘𝑇_𝑛=1𝑤^+𝑚=(𝑛),𝑤^+𝑚(𝑛)_𝑇_𝑛=1𝐺′𝑢_𝑚≤‖‖𝑤_+𝑚(𝑛),𝑤^+𝑚(𝑛)+⟨𝐽′𝑢_𝑚,𝑤^+𝑚⟩‖‖+^𝑇_𝑛=1𝜖||𝑢_𝑚≤‖‖𝑤(𝑛)||+𝑏||𝑤^+𝑚(𝑛)_+𝑚||‖‖+𝜖√𝜆₁𝜆_𝑘+1‖‖𝑢_𝑚‖‖‖‖𝑤^+𝑚‖‖+𝑏√𝑇√𝜆_𝑘+1∶=𝑐₂‖‖𝑤^+𝑚‖‖‖‖𝑤^+𝑚‖‖+𝑐₃‖‖𝑢_𝑚‖‖‖‖𝑤^+𝑚‖‖,(3.5) where 𝑐₁=1−𝜆^𝑘𝜆_𝑘+1>0,𝑐₂=1+𝑏√𝑇√𝜆_𝑘+1,𝑐₃=𝜖√𝜆₁𝜆_𝑘+1.(3.6) And since 𝜖>0 is arbitrary, we have ‖‖𝑤^+𝑚‖‖‖‖𝑢_𝑚‖‖⟶0as𝑚⟶∞.(3.7) Similarly, let 𝜑=𝑤^−𝑚 in (3.3), by (2.8) and (3.4) we get −𝑐₄‖‖𝑤^−𝑚‖‖²=∶1−𝜆^𝑘𝜆_𝑘−1‖‖𝑤^−𝑚‖‖²≥‖‖𝑤^−𝑚‖‖²−𝜆_𝑘𝑇_𝑛=1𝑤^−𝑚=(𝑛),𝑤^−𝑚(𝑛)_𝑇_𝑛=1𝐺^𝑢_𝑚≥−‖‖𝑤_−𝑚(𝑛),𝑤^−𝑚(𝑛)+⟨𝐽^𝑢_𝑚,𝑤^−𝑚⟩‖‖−^𝑇_𝑛=1𝜖||𝑢_𝑚≥−‖‖𝑤_−𝑚(𝑛)||+𝑏||𝑤^−𝑚(𝑛)||‖‖−𝜖𝜆₁‖‖𝑢_𝑚‖‖‖‖𝑤^−𝑚‖‖−𝑏√𝑇√𝜆₁∶=−𝑐₅‖‖𝑤^−𝑚‖‖‖‖𝑤^−𝑚‖‖−𝑐₆‖‖𝑢_𝑚‖‖‖‖𝑤^−𝑚‖‖,(3.8) where 𝑐₄=𝜆^𝑘𝜆_𝑘−1−1>0,𝑐₅=1+𝑏√𝑇√𝜆₁,𝑐₆=𝜖𝜆₁.(3.9) And hence we also have ‖‖𝑤^−𝑚‖‖‖‖𝑢_𝑚‖‖⟶0as𝑚⟶∞.(3.10) By (3.7) and (3.10), we have ‖‖𝑤_𝑚‖‖‖‖𝑢_𝑚‖‖⟶0,‖‖𝑣^𝑚‖‖‖‖𝑢_𝑚‖‖⟶1as𝑚⟶∞.(3.11) By (𝐺⁺), there exist 𝛿>0 and 𝑀∈ℕ such that ^𝑇_𝑛=1𝐺^𝑢_𝑚(𝑛),𝑣_𝑚(𝑛)≥𝛿,∀𝑚≥𝑀.(3.12) This implies that ⟨𝐽^𝑢_𝑚,𝑣_𝑚⟩=−^𝑇_𝑛=1𝐺^𝑢_𝑚(𝑛),𝑣_𝑚(𝑛)≤−𝛿,∀𝑚≥𝑀,(3.13) and hence ‖‖𝐽^𝑢_𝑚‖‖‖‖𝑢_𝑚‖‖≥‖‖𝐽^𝑢_𝑚‖‖‖‖𝑣_𝑚‖‖≥||⟨𝐽^𝑢_𝑚,𝑣_𝑚⟩||≥𝛿,∀𝑚≥𝑀,(3.14) which is a contradiction to (3.2). Thus {𝑢_𝑚} is bounded. The proof is complete.

Lemma 3.2. Let (1.8) and (1.9) hold. Then (1)𝐶_𝑞(𝐽,∞)≅𝛿_𝑞,𝑘𝐹 provided that (𝐺⁺) holds; (2)𝐶_𝑞(𝐽,∞)≅𝛿_{𝑞,𝑘−1}𝐹 provided that (𝐺⁻) holds.

Proof. We only prove the case (1). Define a bilinear function 𝑎(𝑢,𝑣)=𝜆_𝑘𝑇_𝑛=1(𝑢(𝑛),𝑣(𝑛)),∀𝑢,𝑣∈𝐸.(3.15) Then by (2.7) we have ||𝑎(𝑢,𝑣)||≤𝜆^𝑘𝜆₁‖𝑢‖‖𝑣‖.(3.16) And hence there exists a unique continuous bounded linear operator 𝐾∶𝐸→𝐸 such that ⟨𝐾𝑢,𝑣⟩=𝜆_𝑘𝑇_𝑛=1(𝑢(𝑛),𝑣(𝑛)).(3.17) Since ⟨𝐾𝑢,𝑢⟩∈ℝ for all 𝑢∈𝐸, we can conclude that 𝐾 is a self-adjoint operator and 𝐽(𝑢)=12⟨(𝐼−𝐾)𝑢,𝑢⟩−^𝑇_𝑛=1𝐺(𝑢(𝑛)).(3.18) Then 𝐽 has the form (2.16) with 𝑄(𝑢)=−^𝑇_𝑛=1𝐺(𝑢(𝑛)),(3.19) and (1.11) implies that (2.17) holds. Let 𝐴=𝐼−𝐾. Then ker𝐴=𝑊⁰=span{𝜙_𝑘}. Next we show that (𝐺⁺) implies that the angle condition (𝐴𝐶^−∞) at infinity holds.
If not, then for any 𝑚∈ℕ and each 𝛼_𝑚=1/𝑚, there exists 𝑢_𝑚=𝑣_𝑚+𝑤_𝑚∈𝑊⁰⊕(𝑊⁺⊕𝑊⁻) with 𝑣_𝑚∈𝑊⁰, 𝑤_𝑚∈𝑊⁺⊕𝑊⁻ such that ‖‖𝑢_𝑚‖‖≥𝑚,‖‖𝑤_𝑚‖‖≤1𝑚‖‖𝑢^𝑚⟨𝐽′𝑢_𝑚‖‖,(3.20),𝑣_𝑚⟩>0.(3.21) On the other hand, (3.20) implies ‖‖𝑢_𝑚‖‖⟶∞,‖‖𝑣^𝑚‖‖‖‖𝑢_𝑚‖‖⟶1as𝑚⟶∞.(3.22) Thus, by (𝐺⁺) there exist 𝛿>0 and 𝑀∈ℕ such that ^𝑇_𝑛=1𝐺^𝑢_𝑚(𝑛),𝑣_𝑚(𝑛)≥𝛿,∀𝑚≥𝑀.(3.23) Therefore, ⟨𝐽^𝑢_𝑚,𝑣_𝑚⟩=−^𝑇_𝑛=1𝐺^𝑢_𝑚(𝑛),𝑣_𝑚(𝑛)≤−𝛿,∀𝑚≥𝑀,(3.24) which is a contradiction to (3.21). Consequently (𝐴𝐶^−∞) holds and by Lemma 3.1 and Proposition 2.5, 𝐶_𝑞(𝐽,∞)=𝛿_𝑞,𝑘𝐹. Similarly, we can prove that (2) holds.

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

Lemma 3.3. Let 𝑉^+(𝑡)=𝑉^0,(𝑡),𝑡𝑡≤≥00,,𝑉_−(𝑡)=𝑉^0,(𝑡),𝑡𝑡≥≤00,,(3.25) and 𝑉^±(𝑡)=∫^𝑡0𝑉′^±(𝑠)𝑑𝑠. If lim_|𝑡|→∞𝑉^(𝑡)𝑡=𝛼≠𝜆¹,(3.26) then the functional 𝐽^±(𝑢)=12^𝑇_𝑛=0||Δ𝑢(𝑛)||²−^𝑇_𝑛=1𝑉^±(𝑢(𝑛))(3.27) satisfies the (PS) condition.

Proof. We only prove the case (𝐽⁺). Let {𝑢_𝑚}⊂𝐸 such that 𝐽𝑢_𝑚⟶𝑐∈ℝ,𝐽′⁺𝑢_𝑚⟶0(3.28) as 𝑚→∞. Since 𝐸 is a finite-dimensional space, it suffices to show that {𝑢_𝑚} is bounded in 𝐸. Suppose that {𝑢_𝑚} is unbounded. Passing to a subsequence we may assume that ‖𝑢_𝑚‖→∞ and for each 𝑛, either |𝑢_𝑚(𝑛)|→∞ or {𝑢_𝑚(𝑛)} is bounded.
Noticing that for all 𝜑∈𝐸, ⟨𝐽′⁺𝑢_𝑚,𝜑⟩=⟨𝑢_𝑚,𝜑⟩−^𝑇_𝑛=1𝑉′⁺𝑢_𝑚(𝑛),𝜑(𝑛).(3.29)
Denote 𝑤_𝑚∶=𝑢_𝑚/‖𝑢_𝑚‖, for a subsequence, 𝑤_𝑚 converges to some 𝑤 with ‖𝑤‖=1. By (3.29), we have ⟨𝐽′⁺𝑢_𝑚,𝜑⟩‖‖𝑢_𝑚‖‖=⟨𝑤^𝑚,𝜑⟩−^𝑇_𝑛=1𝑉′⁺𝑢_𝑚(𝑛)‖‖𝑢_𝑚‖‖,𝜑(𝑛).(3.30) If |𝑢_𝑚(𝑛)|→∞, then lim_𝑚→∞𝑉′⁺𝑢_𝑚(𝑛)𝑢_𝑚(𝑛)𝑤^𝑚(𝑛)=𝛼𝑤⁺(𝑛),(3.31) where 𝑤⁺(𝑛)=max{𝑤(𝑛),0} with 𝑛∈ℤ(1,𝑇). If {𝑢_𝑚(𝑛)} is bounded, then lim_𝑚→∞𝑉′⁺𝑢_𝑚(𝑛)‖‖𝑢_𝑚‖‖=0,𝑤(𝑛)=0.(3.32) Since 𝑤≠0, there is an 𝑛 for which |𝑢_𝑚(𝑛)|→∞. So passing to the limit in (3.30), we have ^𝑇_𝑛=0(Δ𝑤(𝑛),Δ𝜑(𝑛))−𝛼^𝑇_𝑛=1𝑤⁺(𝑛),𝜑(𝑛)=0.(3.33) This implies that 𝑤≠0 satisfies Δ²𝑤(𝑛−1𝑤)(+0)𝛼=𝑤⁺0(=𝑛)𝑤=(0𝑇,+𝑛1)∈.ℤ(1,𝑇),(3.34)
Now, we claim that 𝑤(𝑛)>0,∀𝑛∈ℤ(1,𝑇).(3.35)
In fact, let 𝑤𝑛₀=min{𝑤(𝑛)∶𝑛∈ℤ(1,𝑇)}.(3.36) We only need to prove 𝑤(𝑛₀)>0. If not, assume that 𝑤(𝑛₀)≤0. Then by (3.34), we have Δ²𝑤(𝑛₀−1)=0 and hence 𝑤(𝑛₀−1)=𝑤(𝑛₀)=𝑤(𝑛₀+1). By induction, it is easy to get 𝑤(𝑛)=0 for all 𝑛∈ℤ(1,𝑇) which is a contradiction to 𝑤≠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 𝜆₁ is positive, which is a contradiction to 𝛼≠𝜆₁. The proof is complete.

Lemma 3.4. Under the conditions of Theorem 1.3, the functional 𝐽⁺ has a critical point 𝑢⁺>0 and 𝐶_𝑞(𝐽⁺,𝑢⁺)≅𝛿_𝑞,1𝐹; the functional 𝐽⁻ has a critical point 𝑢⁻<0 and 𝐶_𝑞(𝐽⁻,𝑢⁻)≅𝛿_𝑞,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 𝑢⁺. In fact, 𝐽⁺∈𝐶¹(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 𝑉^(0)<𝜌₂<𝜆₁ such that 𝑉(𝑡)≤12𝜌²𝑡²(3.37) for |𝑡|≤𝜌₁. So, for all 𝑢∈𝐸, if ‖𝑢‖≤√𝜆₁𝜌₁, then for each 𝑛∈ℤ(1,𝑇), |𝑢(𝑛)|≤𝜌₁ and 𝐽⁺(𝑢)=12‖𝑢‖²−^𝑇_𝑛=1=1𝑉⁺(𝑢(𝑛))2‖𝑢‖²≥1−_𝑛∈𝑁1𝑉(𝑢(𝑛))2‖𝑢‖²−12𝜌²_𝑛∈𝑁1≥1(𝑢(𝑛),𝑢(𝑛))2‖𝑢‖²−12𝜌^2𝑇_𝑛=1≥1(𝑢(𝑛),𝑢(𝑛))2‖𝑢‖²−12𝜆𝜌²₁‖𝑢‖²,(3.38) where 𝑁₁={𝑛∈ℤ(1,𝑇)∣𝑢(𝑛)≥0}. Let 𝜌=√𝜆₁𝜌₁,𝑎=121−𝜆𝜌²₁𝜌².(3.39) Then 𝐽⁺(𝑢)|_𝜕𝐵𝜌≥𝑎>0 and hence (𝐽1) holds. For (𝐽2), by 𝑉^(∞)=𝜆_𝑘∈(𝜆_𝑘−1,𝜆_𝑘+1), we claim that there exist 𝛾>𝜆_𝑘−1(≥𝜆₁), 𝑏∈ℝ such that 𝑉(𝑡)≥𝛾2𝑡²+𝑏,∀𝑡∈ℝ.(3.40)
In fact, by assumption (1.9), there exist 𝑀>0 and 𝑏₁∈ℝ such that 𝑉(𝑡)≥(𝛾/2)𝑡²+𝑏₁ for |𝑡|≥𝑀. Meanwhile, there exists 𝑏₂∈ℝ such that 𝑉(𝑡)−(𝛾/2)𝑡²≥𝑏₂ for |𝑡|≤𝑀 by virtue of the continuity of 𝑉. Let 𝑏=min{𝑏₁,𝑏₂}, we get the conclusion.
Thus, if we choose 𝑒∈span{𝜙₁} with 𝑒>0 and ‖𝑒‖=1, then 𝐽⁺(𝑡𝑒)=𝑡²2−^𝑇_𝑛=1≤𝑡𝑉(𝑡𝑒(𝑛))₂2−𝛾𝑡²=𝑡2(𝑒,𝑒)−𝑏𝑇₂2−𝛾𝑡²2𝜆₁−𝑏𝑇⟶−∞(3.41) as 0<𝑡→+∞. Thus, we can choose a constant 𝑡 large enough with 𝑡>𝜌 and 𝑢₀=𝑡𝑒∈𝐸 such that 𝐽⁺(𝑢₀)≤0. (𝐽2) holds.
Therefore, by Theorem 2.7, 𝐽⁺ has a critical point 𝑢⁺≠0 and similar to the proof of Lemma 3.3, we can prove that 𝑢⁺>0. So 𝑢⁺ is also a critical point of 𝐽. In the following we compute the critical group 𝐶_𝑞(𝐽⁺,𝑢⁺) by using Theorem 2.9.
Assume that ⟨𝐽^𝑢⁺𝑣,𝑣⟩=⟨𝑣,𝑣⟩−^𝑇_𝑛=1𝑉^𝑢⁺(𝑛)𝑣(𝑛),𝑣(𝑛)≥0,∀𝑣∈𝐸,(3.42) and that there exists 𝑣₀≢0 such that ⟨𝐽^𝑢⁺𝑣₀,𝑣⟩=0,∀𝑣∈𝐸.(3.43) This implies that 𝑣₀ satisfies Δ²𝑣₀(𝑛−1)+𝑣𝑉^𝑢⁺(𝑛)𝑣₀(𝑛)=0,𝑛∈ℤ(1,𝑇),₀(0)=𝑣₀(𝑇+1)=0.(3.44) Hence the eigenvalue problem Δ²𝑣(𝑛−1)+𝜆𝑉𝑣(^0)𝑢=⁺(𝑣𝑛)(𝑇𝑣+(𝑛1))==00,𝑛∈ℤ(1,𝑇),(3.45) has an eigenvalue 𝜆=1. (𝑉1) implies that 1 must be a simple eigenvalue; see [4]. So, dimker(𝐽^(𝑢₀))=1. Since 𝐸 is a finite-dimensional Hilbert space, the Morse index of 𝑢⁺ must be finite and 𝐽^(𝑢⁺) must be a Fredholm operator. By Theorem 2.9, 𝐶_𝑞(𝐽⁺,𝑢⁺)≅𝛿_𝑞,1𝐹. The proof is complete.

Remark 3.5. We can choose the neighborhood 𝑈 of 𝑢⁺ such that 𝑢>0 for all 𝑢∈𝑈. Therefore, 𝐶_𝑞𝐽,𝑢⁺≅𝐶_𝑞𝐽⁺,𝑢⁺≅𝛿_𝑞,1𝐹.(3.46) Similarly, 𝐶_𝑞(𝐽,𝑢⁻)≅𝐶_𝑞(𝐽⁻,𝑢⁻)≅𝛿_𝑞,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, 𝐶_𝑞(𝐽,∞)≅𝛿_𝑞,𝑘𝐹.(3.48) Hence by Proposition 2.4 the functional 𝐽 has a critical point 𝑢₁ satisfying 𝐶_𝑘𝐽,𝑢₁≇0.(3.49) Since ⟨𝐽^(0)𝑢,𝑢⟩≥1−𝑉^(0)𝜆₁‖𝑢‖²,(3.50) by (𝑉2) and 𝐽(0)=𝐽^(0)=0, we see that 0 is a local minimum of 𝐽. Hence 𝐶_𝑞(𝐽,0)≅𝛿_𝑞,0𝐹.(3.51) By Remark 3.5, (3.49), (3.51), and 𝑘≥2 we get that 𝑢⁺, 𝑢⁻, and 𝑢₁ are three nonzero critical points of 𝐽 with 𝑢⁺>0 and 𝑢⁻<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 Δ²𝑢(𝑛−1)+𝑢𝑉(0^)(𝑢=(𝑛0)=)=𝑢0(6,),𝑛∈ℤ(1,5),(4.1) where 𝑉∈𝐶²(ℝ,ℝ) is defined as follows: 𝑉(𝑡)=⎧⎪⎨⎪⎩110𝑡²1,|𝑡|≤1,2𝑡²+34𝑡^4/3astrictlyco,nvexfunction,o|𝑡t|h≥er1w0i,se.(4.2) It is easy to verify that 𝑉 satisfies (1.8), (1.9), (1.11), (𝑉1), and (𝑉2) with 𝑘=2. To verify the condition (𝐺⁺), note that 𝐺^(𝑡)=𝑡^1/3 for |𝑡|≥10, we claim that ⁵_𝑛=1𝐺^𝑢_𝑚(𝑛),𝑣_𝑚(𝑛)⟶+∞,as𝑚⟶∞(4.3) which implies that (𝐺⁺) holds.
To this end, for any constant 𝑟>1, we introduce another norm in 𝐸(𝑇=5) as follows: ‖𝑢‖_𝑟=⁵_𝑛=1||𝑢(𝑛)||^𝑟^1/𝑟,∀𝑢∈𝐸.(4.4) Since 𝐸 is finite dimensional, there exist two constants 𝐶₂≥𝐶₁>0 such that 𝐶₁‖𝑢‖≤‖𝑢‖_𝑟≤𝐶₂‖𝑢‖,∀𝑢∈𝐸.(4.5)
Now, by (𝐺⁺), for any 𝜖 small enough, it is easy to see that ‖‖𝑤_𝑚‖‖≤𝜖‖‖𝑢_𝑚‖‖(4.6) holds for 𝑚 large enough.
Set Ω₁=𝑛∈ℤ(1,5)∶||𝑢_𝑚(𝑛)||≥10,Ω₂=ℤ(1,5)⧵Ω₁.(4.7) Since ‖𝑢_𝑚‖→∞, Ω₁≠∅, for 𝑚 large enough. And for 𝑚 large enough, we have ⁵_𝑛=1𝐺^𝑢_𝑚(𝑛),𝑣_𝑚(𝑛)=−_𝑛∈Ω1𝑢^𝑚1/3(𝑛),𝑣_𝑚(𝑛)+_𝑛∈Ω2𝐺^𝑢_𝑚(𝑛),𝑢_𝑚(𝑛)_𝑛∈Ω2≥𝐺′𝑢_𝑚(𝑛),𝑤_𝑚(𝑛)_𝑛∈Ω1=_𝑛∈Ω1𝑢^𝑚1/3(𝑛),𝑣_𝑚(𝑛)−𝑐−𝑐𝜖‖‖𝑢_𝑚‖‖𝑢^𝑚1/3(𝑛),𝑢_𝑚(𝑛)−_𝑛∈Ω1𝑢^𝑚1/3(𝑛),𝑤_𝑚(𝑛)−𝑐−𝑐𝜖‖‖𝑢_𝑚‖‖.(4.8) Here and below we denote by 𝑐 various positive constants. Since _𝑛_∈Ω1𝑢^𝑚1/3(𝑛),𝑢_𝑚(𝑛)=⁵_𝑛=1𝑢^𝑚1/3=‖‖𝑢(𝑛),𝑢_𝑚(𝑛)−_𝑛∈Ω2𝑢^𝑚1/3(𝑛),𝑢_𝑚(𝑛)_𝑚‖‖^4/34/3−𝑐,_𝑛∈Ω1𝑢^𝑚1/3(𝑛),𝑤_𝑚(𝑛)≤≤⁵_𝑛=1||𝑢_𝑚(𝑛)||^1/3||𝑤_𝑚(𝑛)₅||_𝑛=1||𝑢_𝑚(𝑛)||^4/3^1/4⁵_𝑛=1||𝑤_𝑚(𝑛)||^4/3^3/4=‖‖𝑢_𝑚‖‖^1/34/3‖‖𝑤_𝑚‖‖_4/3≤𝑐𝜖‖‖𝑢_𝑚‖‖^4/34/3.(4.9) Hence ⁵_𝑛=1𝐺^𝑢_𝑚(𝑛),𝑣_𝑚(𝑛)≥(1−𝑐𝜖)‖‖𝑢_𝑚‖‖^4/34/3−𝑐𝜖‖‖𝑢_𝑚‖‖−𝑐.(4.10) Since 𝜖 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

Δ²𝑢(𝑛−1)+𝑉′(𝑢(𝑛))=0,𝑛∈ℤ(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 Δ²𝑢(𝑛−1)+𝑢(𝑉0)^(=𝑢(0𝑛=))Δ=𝑢0(,𝑇)𝑛.∈ℤ(1,𝑇),(4.12) Let 𝐸=𝑢∶𝑢={𝑢(𝑛)}^𝑇+1𝑛=0with𝑢(0)=0=Δ𝑢(𝑇).(4.13) Then 𝐸 is a 𝑇-dimensional Hilbert space with the inner product ⟨𝑢,𝑣⟩=^𝑇_𝑛=0(Δ𝑢(𝑛),Δ𝑣(𝑛)).(4.14) Define the functional 𝐽 on 𝐸 by 𝐽(𝑢)=^𝑇_𝑛=012||Δ𝑢(𝑛)||²−^𝑇_𝑛=1𝑉(𝑢(𝑛)).(4.15) It is easy to see that 𝑢 is a critical point of 𝐽 in 𝐸 if and only if 𝑢 is a solution of the BVP (4.12).
The eigenvalues of the linear BVP Δ²𝑢(𝑛−1)𝑢+(0𝜆)𝑢=(𝑛0)==Δ0,𝑢(𝑇𝑛)∈ℤ(1,𝑇),(4.16) are 𝜆=𝜆_𝑙=4sin²𝑙𝜋2(2𝑇+1),𝑙=1,2,…,𝑇,(4.17) and the corresponding eigenfunctions are 𝜙_𝑙(𝑛)=sin𝑙𝜋𝑛2𝑇+1,𝑙=1,2,…,𝑇.(4.18) Hence, 𝜆_𝑙≠0 for all 𝑙∈ℤ(1,𝑇) and 𝜙₁(𝑛)>0 for all 𝑛∈ℤ(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:

Δ²𝑢𝑢(𝑛(0−)1=)𝑢+(𝑇𝑉′)(,𝑢(𝑛))Δ=𝑢(00,)=𝑛Δ∈𝑢ℤ(𝑇()1.,𝑇),(4.19) It is easy to verify that the variational functional of the BVP (4.19) is

𝐽(𝑢)=^𝑇_𝑛=10𝑥0200𝑑12||Δ𝑢(𝑛)||²−𝑉(𝑢(𝑛)),∀𝑢∈𝐸₁,(4.20) where

𝐸₁=𝑢∶𝑢={𝑢(𝑛)}^𝑇+1𝑛=0with𝑢(0)=𝑢(𝑇),Δ𝑢(0)=Δ𝑢(𝑇+1).(4.21) But, 𝜆=0 is an eigenvalue of the linear BVP:

Δ²𝑢𝑢((0𝑛)−=1𝑢)(+𝑇𝜆)𝑢,(𝑛)=Δ𝑢0(,0)𝑛=∈Δℤ𝑢((𝑇1,)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).