Infinitely Many Homoclinic Solutions for Second Order Nonlinear Difference Equations with 𝑝-Laplacian

Abstract

We employ Nehari manifold methods and critical point theory to study the existence of nontrivial homoclinic solutions of discrete 𝑝-Laplacian equations with a coercive weight function and superlinear nonlinearity. Without assuming the classical Ambrosetti-Rabinowitz condition and without any periodicity assumptions, we prove the existence and multiplicity results of the equations.

1. Introduction

Difference equations represent the discrete counterpart of ordinary differential equations and are usually studied in connection with numerical analysis. As it is well known, the critical point theory is used to deal with the existence of solutions of difference equations. For example, in 2003, Guo and Yu [1] introduced a variational structure associated with second order difference equations; they employ Rabinowitz's saddle point theorem (see [2]) to obtain the existence of 𝑘𝑇-periodic solutions for the 𝑇-periodic system: Δ²𝑢(𝑛−1)+∇_𝑢𝐹[𝑛,𝑢(𝑛)]=0,𝑛∈ℤ. (1) The forward difference operator Δ is defined by Δ𝑢(𝑛)=𝑢(𝑛+1)−𝑢(𝑛). They assume that ∇_𝑢𝐹 is bounded and 𝐹 is coercive with respect to 𝑢 or 𝐹 satisfies a subquadratic Ambrosetti-Rabinowitz condition and a related coercivity condition. In particular, when 𝐹(𝑛,0)=0 for all 𝑛∈ℤ, they prove the existence of nontrivial 𝑘𝑇-periodic solutions of (1). In [3], they assume that 𝐹 satisfies a superquadratic Ambrosetti-Rabinowitz condition and ∇_𝑢𝐹 satisfies a superlinear condition near 𝑢=0 and prove the existence of two nontrivial 𝑇-periodic solutions of (1) by using the similar methods. A survey of those results is given in [4]. In 2004, Zhou et al. [5] consider the case, where the nonlinearity is neither superlinear nor sublinear and generalize the results of [3]. In these papers, the critical point theory is applied to find the periodic solutions of difference equations. The main idea of these papers is to construct a suitable variational structure, so that the critical points of the variational functional correspond to the periodic solutions of the difference equations. Naturally, the critical point theory is also applied to find homoclinic solutions of difference equations; see [6–11] and the reference therein.

In this paper, we consider the following second order nonlinear difference equations with 𝑝-Laplacian: −Δ𝜙_𝑝(Δ𝑢(𝑘−1))+𝑏(𝑘)𝜙_𝑝(𝑢(𝑘))=𝑓(𝑘,𝑢(𝑘)),𝑘∈ℤ, (2) where 𝜙_𝑝(𝑡)=|𝑡|^𝑝−2𝑡 for all 𝑡∈ℝ, 𝑝>1. 𝑏∶ℤ→ℝ is a positive and coercive weight function and 𝑓(𝑘,𝑢)∶ℤ×ℝ→ℝ is a continuous function on 𝑢. The forward difference operator Δ is defined by Δ𝑢(𝑘−1)=𝑢(𝑘)−𝑢(𝑘−1),∀𝑘∈ℤ. (3) As usual, ℤ and ℝ denote the set of all integers and real numbers, respectively.

Assume further that 𝑓(𝑘,0)=0; then 𝑢(𝑘)≡0 is a solution of (2), which is called the trivial solution. As usual, we say that a solution 𝑢={𝑢(𝑘)} of (2) is homoclinic (to 0): if lim_|𝑘|→∞𝑢(𝑘)=0. (4) In addition, we are interested in the existence of nontrivial homoclinic solution for (2), that is, solutions that are not equal to 0 identically. In this paper, we also obtain infinitely many homoclinic solutions of (2) for case, where 𝑓 is odd in 𝑢.

Moreover, we may regard (2) as being a discrete analogue of the following second order differential equation: −𝜙_𝑝𝑥^′(𝑡)^′+𝑏(𝑡)𝜙_𝑝(𝑥(𝑡))=𝑓(𝑡,𝑥(𝑡)),𝑡∈ℝ. (5)

The study of homoclinic solutions for (2) in case 𝑝=2 has been motivated in part by searching standing waves for the nonlinear discrete Schrödinger equation: 𝑖𝜓̇_𝑛+Δ²𝜓_𝑛−𝑣_𝑛𝜓_𝑛+𝑓𝑛,𝜓_𝑛=0,𝑛∈ℤ, (6) namely, solutions of the form 𝜓_𝑛=𝑢_𝑛𝑒^{−𝑖𝜔𝑡}. Periodic assumptions on (6) can be found in [6, 7]. Without any periodic assumptions, the existence and multiplicity of standing wave solutions of (6) are obtained in [8, 9]. We are going to extend the approach of [8] to nonlinear discrete 𝑝-Laplacian equations.

Throughout this paper, we always suppose that the following conditions hold:(𝐵)function 𝑏∶ℤ→ℝ satisfies 𝑏(𝑘)≥𝑏₀>0 for all 𝑘∈ℤ and lim_{|𝑘|→+∞}𝑏(𝑘)=+∞. (7)(𝑓₁)𝑓∈𝐶(ℤ×ℝ,ℝ) and there exist 𝑎>0, 𝑞∈(𝑝,∞), such that 𝑓(𝑘,𝑢)≤𝑎1+|𝑢|^𝑞−1,∀𝑘∈ℤ,𝑢∈ℝ. (8)(𝑓₂)lim_|𝑢|→0𝑓(𝑘,𝑢)/|𝑢|^𝑝−1=0 uniformly for 𝑘∈ℤ.(𝑓₃)lim_|𝑢|→∞𝐹(𝑘,𝑢)/|𝑢|^𝑝=+∞ uniformly for 𝑘∈ℤ, where 𝐹(𝑘,𝑢) is the primitive function of 𝑓(𝑘,𝑢); that is, 𝐹(𝑘,𝑢)=^𝑢₀𝑓(𝑘,𝑠)𝑑𝑠. (9)(𝑓₄)𝑢↦𝑓(𝑘,𝑢)/|𝑢|^𝑝−1 is strictly increasing on (−∞,0) and (0,∞).

In many studies of 𝑝-Laplacian equations, the following classical Ambrosetti-Rabinowitz superlinear condition ([12, 13]) is assumed: 0<𝜇𝐹(𝑘,𝑢)≤𝑓(𝑘,𝑢)𝑢,forsome𝜇>𝑝,𝑢≠0. (10) It is easy to see that (10) implies 𝐹(𝑘,𝑢)≥𝐶|𝑢|^𝜇, for some constants 𝐶>0 and |𝑢|≥1.

In this paper, instead of (10), we assume the 𝑝-superlinear condition (𝑓₃). It is easy to see that (10) implies (𝑓₃). For example, the 𝑝-superlinear function, 𝑓(𝑘,𝑢)=|𝑢|^𝑝−2𝑢ln(1+|𝑢|), (11) does not satisfy (10). However, it satisfies the condition (𝑓₁)--(𝑓₃). A crucial role that (10) plays is to ensure the boundedness of Palais-Smale sequences. This is very crucial in applying the critical point theory.

The rest of the paper is organized as follows. In Section 2, we establish the variational framework associated with (2) and then present the main results of this paper. Section 3 is devoted to prove some useful lemmas, and in Section 4 we prove the main result.

2. Preliminaries

We will establish the corresponding variational framework associated with (2).

Consider the real sequence spaces 𝑙^𝑝≡𝑙^𝑝(ℤ)=𝑢={𝑢(𝑘)}_𝑘∈ℤ∶∀𝑘∈ℤ,𝑢(𝑘)∈ℝ,‖𝑢‖_𝑙^𝑝=_𝑘∈ℤ|𝑢(𝑘)|^𝑝1/𝑝<∞. (12) Then the following embedding between 𝑙^𝑝 spaces holds: 𝑙^𝑞⊂𝑙^𝑝,‖𝑢‖_𝑙^𝑝≤‖𝑢‖_𝑙^𝑞,1≤𝑞≤𝑝≤∞. (13)

Define the space 𝐸∶=𝑢∈𝑙^𝑝∶_𝑘∈ℤ|Δ𝑢(𝑘−1)|^𝑝+𝑏(𝑘)|𝑢(𝑘)|^𝑝<∞. (14) Then 𝐸 is a Hilbert space equipped with the norm ‖𝑢‖^𝑝=_𝑘∈ℤ|Δ𝑢(𝑘−1)|^𝑝+𝑏(𝑘)|𝑢(𝑘)|^𝑝. (15)|⋅| is the usual absolute value in ℝ.

Now we consider the variational functional 𝐽 defined on 𝐸 by 𝐽(𝑢)=1𝑝_𝑘∈ℤ|Δ𝑢(𝑘−1)|^𝑝+𝑏(𝑘)|𝑢(𝑘)|^𝑝−_𝑘∈ℤ𝐹((𝑘=1𝑘,𝑢))𝑝‖𝑢‖^𝑝−_𝑘∈ℤ𝐹(𝑘,𝑢(𝑘)). (16) Then 𝐽∈𝐶¹(𝐸,ℝ), for all 𝑣∈𝐸, 𝐽^′(𝑢),𝑣=_𝑘∈ℤ𝜙_𝑝((()(𝑘)𝜙Δ𝑢𝑘−1))Δ𝑣𝑘−1+𝑏_𝑝(𝑢(𝑘))𝑣(𝑘)−_𝑘∈ℤ𝑓((𝑘𝑣(𝑘),(𝑢)𝑘,𝑢))𝜕𝐽𝜕𝑢(𝑘)=−Δ𝜙_𝑝(((𝑘)𝜙Δ𝑢𝑘−1))+𝑏_𝑝(𝑢(𝑘))−𝑓(𝑘,𝑢(𝑘)),𝑘∈ℤ. (17) Thus, 𝑢 is a critical point of 𝐽 on 𝐸 only if 𝑢 is homoclinic solutions of (2). We have reduced the problem of finding homoclinic solutions of (2) to that of seeking critical points of the functional 𝐽 on 𝐸. This means that functional 𝐽 is just the variational framework of (2).

The following lemma plays an important role in this paper; it was established in [11].

Lemma 1. If 𝑣 satisfies the condition (𝐵), for any 𝑞>1, then the embedding map from 𝐸 into 𝑙^𝑞(ℤ) is compact.

The main result is as follows.

Theorem 2. Suppose conditions (𝐵),(𝑓₁)--(𝑓₄) are satisfied. Then we have the following conclusions.(1)Equation (2) has a nontrivial ground state homoclinic solution, that is, homoclinic solutions corresponding to the least positive critical value of the variational functional.(2)If 𝑓(𝑘,𝑢) is odd in 𝑢 for each 𝑘∈ℤ, (2) has infinitely many pairs of homoclinic solutions ±𝑢^(𝑘) in 𝐸.

To prove the multiplicity results, we need the following lemma.

Lemma 3 (see [14]). Let 𝑆={𝑤∈𝐸∶‖𝑤‖=1}. If 𝐸 is an infinite-dimensional Hilbert space, Φ∈𝐶¹(𝑆,ℝ) is even and bounded below and satisfies the Palais-Smale condition. Then Φ has infinitely many pairs of critical points.

3. Some Useful Lemmas

We define the Nehari manifold: 𝒩=𝑢∈𝐸⧵{0}∶𝐽^′(𝑢)𝑢=0. (18)

To prove the main results, we need some lemmas.

Lemma 4. Suppose conditions (𝐵),(𝑓₁)--(𝑓₄) are satisfied. Then, for each 𝑤∈𝐸⧵{0}, there exists a unique 𝑠_𝑤>0 such that 𝑠_𝑤𝑤∈𝒩.

Proof. Let 𝐼(𝑢)=∑_𝑘∈ℤ𝐹(𝑘,𝑢(𝑘)). By (𝑓₂), we have 𝐼^′(𝑢)=𝑜‖𝑢‖^𝑝−1as𝑢⟶0. (19) From (𝑓₄), for all 𝑢≠0 and 𝑠>0, we have 𝐼𝑠⟼^′(𝑠𝑢)𝑢𝑠^𝑝−1isstrictlyincreasing. (20) Let 𝑊⊂𝐸⧵{0} be a weakly compact subset and 𝑠>0; we claim that 𝐼(𝑠𝑢)𝑠^𝑝⟶∞uniformlyfor𝑢on𝑊,as𝑠⟶∞. (21) Indeed, let {𝑢_𝑛}⊂𝑊. It suffices to show that if𝑠_𝑛𝐼𝑠⟶∞,^𝑛𝑢_𝑛𝑠_𝑛^𝑝⟶∞, (22) as 𝑛→∞. Passing to a subsequence if necessary, 𝑢_𝑛⇀𝑢∈𝐸⧵{0} and 𝑢_𝑛(𝑘)→𝑢(𝑘) for every 𝑘, as 𝑛→∞.
Note that, from (𝑓₂) and (𝑓₄), it is easy to get that 𝐹(𝑘,𝑢)>0,∀𝑢≠0. (23) Since |𝑠_𝑛𝑢_𝑛(𝑘)|→∞ and 𝑢_𝑛≠0, by (𝑓₃) and (23), we have 𝐼𝑠_𝑛𝑢_𝑛𝑠_𝑛^𝑝=_𝑘∈ℤ𝐹𝑘,𝑠_𝑛𝑢_𝑛(𝑘)𝑠_𝑛𝑢_𝑛(𝑘)^𝑝𝑢_𝑛(𝑘)^𝑝⟶∞as𝑛⟶∞. (24) Therefore, (21) holds.
Let 𝑔(𝑠)∶=𝐽(𝑠𝑤),𝑠>0. Then 𝑔^′(𝑠)=𝐽^′(𝑠𝑤)𝑤=𝑠^𝑝−1‖𝑤‖^𝑝−𝑠^1−𝑝𝐼^′(𝑠𝑤)𝑤, (25) from (19)–(21); then there exists a unique 𝑠_𝑤, such that 𝑔^′(𝑠)>0 whenever 0<𝑠<𝑠_𝑤, 𝑔^′(𝑠)<0 whenever 𝑠>𝑠_𝑤, and 𝑔^′(𝑠_𝑤)=𝐽^′(𝑠_𝑤𝑤)𝑤=0. So 𝑠_𝑤𝑤∈𝒩.

Lemma 5. Suppose conditions (𝐵),(𝑓₁)--(𝑓₄) are satisfied. Then 𝐽 satisfies the Palais-Smale condition on 𝒩.

Proof. Let {𝑢_𝑛}⊂𝒩 be a sequence such that 𝐽(𝑢_𝑛)≤𝑑 for some 𝑑>0 and 𝐽^′(𝑢_𝑛)→0 as 𝑛→∞.
Firstly, we prove that {𝑢_𝑛} is bounded. In fact, if not, we may assume by contradiction that ‖𝑢_𝑛‖→∞ as 𝑛→∞. Let 𝑣_𝑛=𝑢_𝑛/‖𝑢_𝑛‖. Then there exists a subsequence, still denoted by the same notation, such that 𝑣_𝑛⇀𝑣  in  𝐸 as 𝑛→∞.
Suppose 𝑣=0. For every 𝑠>0, from Lemma 4, we have 𝑑≥𝐽𝑢_𝑛≥𝐽𝑠𝑣_𝑛=1𝑝𝑠^𝑝𝑣_𝑛^𝑝−𝐼𝑠𝑣_𝑛⟶1𝑝𝑠^𝑝. (26) This is a contradiction if 𝑠≥^𝑝𝑝𝑑. Therefore, 𝑣≠0.
According to (21), we have 𝐽𝑢0≤^𝑛𝑢_𝑛^𝑝=1𝑝−𝐼𝑢_𝑛𝑣_𝑛𝑢_𝑛^𝑝⟶−∞,𝑛⟶∞, (27) a contradiction again. Thus, {𝑢_𝑛} is bounded.
Finally, we show that there exists a convergent subsequence of {𝑢_𝑛}. Actually, there exists a subsequence, still denoted by the same notation, such that 𝑢_𝑛⇀𝑢. By Lemma 1, for any 𝑞>1, then 𝑢_𝑛⟶𝑢in𝑙^𝑞(ℤ). (28) Note that 𝑢_𝑛−𝑢^𝑝=𝐽^′𝑢_𝑛−𝐽^′(𝑢),𝑢_𝑛−𝑢+_𝑘∈ℤ𝑓𝑘,𝑢_𝑛(𝑘)((𝑘×𝑢−𝑓𝑘,𝑢))_𝑛(𝑘)−𝑢(𝑘). (29) By the weak convergence, the first term on the right hand side of (29) approaches 0 as 𝑘→∞.
By (𝑓₁) and (𝑓₂), it is easy to show that, for any 𝜀>0, there exists 𝑐_𝜀>0, such that 𝑓(𝑘,𝑢)≤𝜀|𝑢|^𝑝−1+𝑐_𝜀|𝑢|^𝑞−1,|𝐹(𝑘,𝑢)|≤𝜀|𝑢|^𝑝+𝑐_𝜀|𝑢|^𝑞. (30) By Hölder's inequality, we have _𝑘∈ℤ𝑓𝑘,𝑢_𝑛(𝑘)−𝑓(𝑘,𝑢(𝑘))𝑢_𝑛(𝑘)(𝑘)≤−𝑢_𝑘∈ℤ𝜀𝑢_𝑛(𝑘)^𝑝−1+|𝑢(𝑘)|^𝑝−1+𝑐_𝜀𝑢_𝑛(𝑘)^𝑞−1+|𝑢(𝑘)|^𝑞−1𝑢_𝑛(𝑘)(𝑘𝑢−𝑢)≤𝜀_𝑛𝑙^𝑝−1_𝑝+‖𝑢‖_𝑙^𝑝−1_𝑝𝑢_𝑛−𝑢_𝑙^𝑝+𝑐_𝜀𝑢_𝑛𝑙^𝑞−1_𝑝+‖𝑢‖_𝑙^𝑞−1_𝑝𝑢_𝑛−𝑢_𝑙^𝑝. (31) Combining (28) and the boundedness of {𝑢^(𝑘)}, the above inequality implies _𝑘∈ℤ𝑓𝑘,𝑢_𝑛(𝑘)−𝑓(𝑘,𝑢(𝑘))𝑢_𝑛(𝑘)−𝑢(𝑘)⟶0as𝑘⟶∞. (32) It follows from (29) that 𝑢_𝑛→𝑢 in  𝐸. This implies that 𝐽 satisfies the Palais-Smale condition.

4. Proof of Main Results

Proof of Theorem 2. (1) Now we need five steps to finish this proof.
Step  1. We claim that 𝒩 is homeomorphic to the unit sphere 𝑆 in 𝐸.
By (19) and (21), 𝑔(𝑠)>0 for 𝑠>0 small and 𝑔(𝑠)<0 for 𝑠>0 large. So 𝑠_𝑤 is a unique maximum of 𝑔(𝑠) and 𝑠_𝑤𝑤 is the unique point on the ray 𝑠↦𝑠𝑤 (𝑠>0) which intersects 𝒩. That is, 𝑢∈𝒩 is the unique maximum of 𝐽 on the ray. Therefore, by Lemma 4, we may define the mapping 𝑚∶𝐸⧵{0}→𝒩 by setting 𝑚(𝑤)∶=𝑠_𝑤𝑤. (33) Next we show that the mapping 𝑚 is continuous. Indeed, suppose 𝑤_𝑛→𝑤≠0. Since 𝑚(𝑡𝑢)=𝑚(𝑢), for each 𝑡>0, we may assume 𝑤_𝑛∈𝑆 for all 𝑛. Write 𝑚(𝑤_𝑛)=𝑠_𝑤𝑛𝑤_𝑛. Then {𝑠_𝑤𝑛} is bounded. If not, 𝑠_𝑤𝑛→∞ as 𝑛→∞.
Note that, by (𝑓₄), for all 𝑢≠0, 1𝑝𝑓()()=1𝑘,𝑢𝑢−𝐹𝑘,𝑢𝑝𝑓(𝑘,𝑢)𝑢−^𝑢₀𝑓()>1𝑘,𝑠𝑑𝑠𝑝𝑓()𝑓()𝑘,𝑢𝑢−𝑘,𝑢𝑢^𝑝−1𝑢₀𝑠^𝑝−1𝑑𝑠=0. (34)
Therefore, for all 𝑢∈𝒩, we have 𝐽(𝑢)(𝑢)−1=𝐽𝑝𝐽^′(𝑢)𝑢=_𝑘∈ℤ1𝑝𝑓(𝑘,𝑢(𝑘))𝑢(𝑘)−𝐹(𝑘,𝑢(𝑘))>0. (35) Combining with (𝑓₃) and Lemma 4, we have 𝐽𝑠0<^𝑤𝑛𝑤𝑠_𝑤𝑛^𝑝=1𝑝‖𝑤‖^𝑝−_𝑘∈ℤ𝐹𝑘,𝑠_𝑤𝑛𝑤(𝑘)𝑠_𝑤𝑛𝑤(𝑘)^𝑝|𝑤(𝑘)|^𝑝⟶−∞,as𝑛⟶∞; (36) this is a contradiction. Therefore, 𝑠_𝑤𝑛→𝑠>0 after passing to a subsequence if needed, since 𝒩 is closed and 𝑚(𝑤_𝑛)=𝑠_𝑤𝑛𝑤_𝑛→𝑠𝑤,𝑠𝑤∈𝒩. Hence 𝑠𝑤=𝑠_𝑤𝑤=𝑚(𝑤) by the uniqueness of 𝑠_𝑤 of Lemma 4. Therefore, 𝑚 is continuous.
Then we define a mapping 𝑚∶𝑆→𝒩 by setting 𝑚∶=𝑚|_𝑆, then 𝑚 is a homeomorphism between 𝑆 and 𝒩, and the inverse of 𝑚 is given by 𝑚⁻¹(𝑢)=𝑢/‖𝑢‖.
Step  2. Now we define the functional Ψ∶𝐸⧵{0}→ℝ and Ψ∶𝑆→ℝ by Ψ(𝑤)∶=𝐽(𝑚(𝑤)),Ψ(𝑤)∶=Ψ|_𝑆. (37) Then we have
Ψ∈𝐶¹(𝐸⧵{0},ℝ) and Ψ∈𝐶¹(𝑆,ℝ). Moreover, Ψ^′(𝑤)‖𝑚(𝑤𝑧=)‖‖𝑤‖𝐽^′(𝑚(𝑤))𝑧∀𝑤,𝑧∈𝐸,𝑤≠0,Ψ^′(𝑤)𝑧=‖𝑚(𝑤)‖𝐽^′(𝑚(𝑤))𝑧∀𝑧∈𝑇_𝑤(𝑆)={𝑣∈𝐸∶(𝑤,𝑣)=0}. (38)(39) In fact, let 𝑤∈𝐸⧵{0} and 𝑧∈𝐸. By Lemma 4 and the mean value theorem, we obtain Ψ(𝑤+𝑡𝑧)−Ψ(𝑤)=𝐽𝑠_{𝑤+𝑡𝑧}(𝑤+𝑡𝑧)−𝐽𝑠_𝑤𝑤≤𝐽𝑠_{𝑤+𝑡𝑧}(𝑤+𝑡𝑧)−𝐽𝑠_{𝑤+𝑡𝑧}(𝑤)=𝐽^′𝑠_{𝑤+𝑡𝑧}𝑤+𝜏_𝑡𝑡𝑧𝑠_{𝑤+𝑡𝑧}𝑡𝑧, (40) where |𝑡| is small enough and 𝜏_𝑡∈(0,1). Similarly, Ψ(𝑤+𝑡𝑧)−Ψ(𝑤)=𝐽𝑠_{𝑤+𝑡𝑧}(𝑤+𝑡𝑧)−𝐽𝑠_𝑤𝑤≥𝐽𝑠_𝑤(𝑤+𝑡𝑧)−𝐽𝑠_𝑤(𝑤)=𝐽^′𝑠_𝑤𝑤+𝜂_𝑡𝑡𝑧𝑠_𝑤𝑡𝑧, (41) where 𝜂_𝑡∈(0,1). From the above, the function 𝑤↦𝑠_𝑤 is continuous, combining these two inequalities that lim_𝑡→0Ψ(𝑤+𝑡𝑧)−Ψ(𝑤)𝑡=𝑠_𝑤𝐽^′𝑠_𝑤𝑤𝑧=‖𝑚(𝑤)‖‖𝑤‖𝐽^′(𝑚(𝑤))𝑧. (42) Hence the Gâteaux derivative of  Ψ is bounded linear in 𝑧 and continuous in 𝑤. It follows that Ψ is a class of 𝐶¹ (see [15], Proposition 1.3) and (38) holds. Note only that, since 𝑤∈𝑆, 𝑚(𝑤)=𝑚(𝑤), so (39) is clear.
Step  3. {𝑤_𝑛} is a Palais-Smale sequence for Ψ if and only if {𝑚(𝑤_𝑛)} is a Palais-Smale sequence for 𝐽.
Let {𝑤_𝑛} be a Palais-Smale sequence for Ψ and let 𝑢_𝑛=𝑚(𝑤_𝑛)∈𝒩. Since for every 𝑤_𝑛∈𝑆 we have an orthogonal splitting, 𝐸=𝑇_𝑤𝑛𝑆⊕ℝ𝑤_𝑛, we have Ψ^′𝑤_𝑛=sup_{𝑧∈𝑇𝑤𝑛𝑆‖𝑧‖=1}Ψ^′𝑤_𝑛𝑧=𝑚𝑤_𝑛sup_{𝑧∈𝑇𝑤𝑛𝑆‖𝑧‖=1}𝐽^′𝑚𝑤_𝑛𝑧=𝑢_𝑛sup_{𝑧∈𝑇𝑤𝑛𝑆‖𝑧‖=1}𝐽^′𝑢_𝑛𝑧. (43) Then Ψ^′𝑤_𝑛≤𝑢_𝑛𝐽^′𝑢_𝑛=𝑢_𝑛sup_{𝑧∈𝑇𝑤𝑛𝑆,𝑡∈ℝ𝑧+𝑡𝑤≠0}𝐽^′𝑢_𝑛(𝑧+𝑡𝑤)‖‖≤𝑢𝑧+𝑡𝑤_𝑛sup_{𝑧∈𝑇𝑤𝑛{0}𝑆⧵}𝐽^′𝑢_𝑛(𝑧)‖𝑧‖=Ψ^′𝑤_𝑛. (44) Therefore Ψ^′𝑤_𝑛=𝑢_𝑛𝐽^′𝑢_𝑛. (45) By (35), for 𝑢_𝑛∈𝒩, 𝐽(𝑢_𝑛)>0, so there exists a constant 𝑐>0 such that 𝐽(𝑢_𝑛)>𝑐. And since 𝑐≤𝐽(𝑢_𝑛)=(1/𝑝)‖𝑢_𝑛‖^𝑝−𝐼(𝑢_𝑛)≤(1/𝑝)‖𝑢_𝑛‖^𝑝, ‖𝑢_𝑛‖≥√^𝑝𝑝𝑐. Together with Lemma 5, √^𝑝𝑝𝑐≤‖𝑢_𝑛‖≤sup_𝑛‖𝑢_𝑛‖<∞. Hence {𝑤_𝑛} is a Palais-Smale sequence for Ψ if and only if {𝑢_𝑛} is a Palais-Smale sequence for 𝐽.
Step  4. By (45), Ψ^′(𝑤)=0 if and only if 𝐽^′(𝑚(𝑤))=0. So 𝑤 is a critical point of Ψ if and only if 𝑚(𝑤) is a nontrivial critical point of 𝐽. Moreover, the corresponding values of Ψ and 𝐽 coincide and inf_𝑆Ψ=inf_𝒩𝐽.
Step  5. Ψ satisfies the Palais-Smale condition.
Let {𝑤_𝑛} be a Palais-Smale sequence for Ψ; then {𝑢_𝑛} is a Palais-Smale sequence for 𝐽 by Step 3, where 𝑢_𝑛∶=𝑚(𝑤_𝑛)∈𝒩. From Lemma 5, 𝑢_𝑛→𝑢 after passing to a subsequence and 𝑤_𝑛→𝑚⁻¹(𝑢), so Ψ satisfies the Palais-Smale condition.
Let {𝑤_𝑛}⊂𝑆 be a minimizing sequence for Ψ. By Ekeland's variational principle we may assume Ψ^′(𝑤_𝑛)→0 as 𝑛→∞, so {𝑤_𝑛} is a Palais-Smale sequence for Ψ. By the Palais-Smale condition, 𝑤_𝑛→𝑤 after passing to a subsequence if needed. Hence 𝑤 is a minimizer for Ψ and therefore a critical point of  Ψ; then 𝑢=𝑚(𝑤) is a critical point of 𝐽 and also is a minimizer for 𝐽. Therefore, 𝑢 is a ground state solution of (2).
(2) If 𝑓(𝑘,𝑢) is odd in 𝑢 for each 𝑘∈ℤ, then 𝐽 is even, so is Ψ. Since inf_𝑆Ψ=inf_𝒩𝐽>0 and Ψ satisfies the Palais-Smale condition, Ψ has infinitely many pairs of critical points by Lemma 3. It follows that (2) has infinitely many pairs of homoclinic solutions ±𝑢_𝑛 in 𝐸.
This completes Theorem 2.

Finally, we exhibit examples to demonstrate the applicability of Theorem 2.

Example 6. Consider the second order difference equation: −Δ²𝑢(𝑘−1)+𝑏(𝑘)𝑢(𝑘)=𝑓(𝑘,𝑢(𝑘)),𝑘∈ℤ, (46) where 𝑏∶ℤ→(0,ℝ) such that lim_{|𝑘|→+∞}𝑏(𝑘)=+∞. Let 𝑓(𝑘,𝑢)=𝑢ln(1+|𝑢|), (47) for all 𝑘∈ℤ. Then it is clear that all conditions of Theorem 2 are satisfied. By Theorem 2, (46) has infinitely many pairs of homoclinic solutions.

Example 7. Consider the 𝑝-Laplacian difference equation: −Δ|Δ𝑢(𝑘−1)|^𝑝−2((𝑘𝑢(𝑘Δ𝑢𝑘−1)+𝑏)|)|^𝑝−2𝑢(𝑘)=𝑓(𝑘,𝑢(𝑘)),𝑘∈ℤ, (48) where 𝑏∶ℤ→(0,ℝ) such that lim_{|𝑘|→+∞}𝑏(𝑘)=+∞. Let 𝑓(𝑘,𝑢)=|𝑢|0,𝑢=0,_𝑝−2𝑢|𝑢||𝑢||𝑢|1−ln,0<≤1,_𝑝−2𝑢(1+ln|𝑢|),|𝑢|>1, (49) for all 𝑘∈ℤ and 𝑝>1. It is easy to verify that 𝑓(𝑘,𝑢) satisfies all conditions in Theorem 2. Therefore, (48) has infinitely many pairs of homoclinic solutions.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the anonymous referee for his/her valuable suggestions. This work is supported by Program for the National Natural Science Foundation of China (no. 11371313) and Biomathematics Laboratory of Yuncheng University (SWSX201302, SWSX201305).