Existence and Multiplicity Results of Homoclinic Solutions for the DNLS Equations with Unbounded Potentials

Abstract

A class of difference equations which include discrete nonlinear Schrödinger equations as special cases are considered. New sufficient conditions of the existence and multiplicity results of homoclinic solutions for the difference equations are obtained by making use of the mountain pass theorem and the fountain theorem, respectively. Recent results in the literature are generalized and greatly improved.

1. Introduction

Assume that 𝑚 is a positive integer. Consider the following difference equation in infinite 𝑚 dimensional lattices, 𝐿𝑢_𝑛+𝑣_𝑛𝑢_𝑛−𝜔𝑢_𝑛=𝜎𝑓𝑛,𝑢_𝑛,𝑛∈ℤ^𝑚, (1.1) where 𝜎=±1, 𝑛=(𝑛₁,𝑛₂,…,𝑛_𝑚)∈ℤ^𝑚, {𝑢_𝑛} is a real valued sequence, 𝜔∈ℝ, 𝐿 is a Jacobi operator [1] given by 𝐿𝑢_𝑛=𝑎_{1(𝑛1,𝑛2,…,𝑛𝑚)}𝑢_{(𝑛1+1,𝑛2,…,𝑛𝑚)}+𝑎_{1(𝑛1−1,𝑛2,…,𝑛𝑚)}𝑢_{(𝑛1−1,𝑛2,…,𝑛𝑚)}+𝑎_{2(𝑛1,𝑛2,…,𝑛𝑚)}𝑢_{(𝑛1,𝑛2+1,…,𝑛𝑚)}+𝑎_{2(𝑛1,𝑛2−1,…,𝑛𝑚)}𝑢_{(𝑛1,𝑛2−1,…,𝑛𝑚)}+⋯+𝑎_{𝑚(𝑛1,𝑛2,…,𝑛𝑚)}𝑢_{(𝑛1,𝑛2,…,𝑛𝑚+1)}+𝑎_{𝑚(𝑛1,𝑛2,…,𝑛𝑚−1)}𝑢_{(𝑛1,𝑛2,…,𝑛𝑚−1)}+𝑏_{(𝑛1,𝑛2,…,𝑛𝑚)}𝑢_{(𝑛1,𝑛2,…,𝑛𝑚)}, (1.2) where {𝑎_𝑖𝑛}(𝑖=1,2,…,𝑚) and {𝑏_𝑛} are real valued and bounded sequences.

We assume that 𝑓(𝑛,0)=0 for 𝑛∈ℤ^𝑚, then 𝑢_𝑛=0 is a solution of (1.1), which is called the trivial solution. As usual, we say that a solution 𝑢={𝑢_𝑛} of (1.1) is homoclinic (to 0) if lim_|𝑛|→∞𝑢_𝑛=0, (1.3) where |𝑛|=|𝑛₁|+|𝑛₂|+⋯+|𝑛_𝑚| is the length of multiindex 𝑛. In addition, if 𝑢_𝑛≢0, then 𝑢 is called a nontrivial homoclinic solution. We are interested in the existence and multiplicity of the nontrivial homoclinic solutions for (1.1). This problem appears when we look for the discrete solitons of the following Discrete Nonlinear Schrödinger (DNLS) equation: 𝑖𝜓̇_𝑛+Δ𝜓_𝑛−𝑣_𝑛𝜓_𝑛+𝜎𝑓𝑛,𝜓_𝑛=0,𝑛∈ℤ^𝑚, (1.4) where Δ𝜓_𝑛=𝜓_{(𝑛1+1,𝑛2,…,𝑛𝑚)}+𝜓_{(𝑛1,𝑛2+1,…,𝑛𝑚)}+⋯+𝜓_{(𝑛1,𝑛2,…,𝑛𝑚+1)}−2𝑚𝜓_{(𝑛1,𝑛2,…,𝑛𝑚)}+𝜓_{(𝑛1−1,𝑛2,…,𝑛𝑚)}+𝜓_{(𝑛1,𝑛2−1,…,𝑛𝑚)}+⋯+𝜓_{(𝑛1,𝑛2,…,𝑛𝑚−1)} (1.5) is the discrete Laplacian in 𝑚 spatial dimension. Moreover, assume that the nonlinearity 𝑓(𝑛,𝑢) is gauge invariant, that is, 𝑓𝑛,𝑒^𝑖𝜃𝑢=𝑒^𝑖𝜃𝑓(𝑛,𝑢),𝜃∈ℝ. (1.6) Since solitons are spatially localized time-periodic solutions and decay to zero at infinity. Thus 𝜓_𝑛 has the form 𝜓_𝑛=𝑢_𝑛𝑒^{−𝑖𝜔𝑡},lim_|𝑛|→∞𝜓_𝑛=0, (1.7) where {𝑢_𝑛} is a real-valued sequence, and 𝜔∈ℝ is the temporal frequency. Then (1.4) becomes −Δ𝑢_𝑛+𝑣_𝑛𝑢_𝑛−𝜔𝑢_𝑛=𝜎𝑓𝑛,𝑢_𝑛,𝑛∈ℤ^𝑚, (1.8) and (1.3) holds. Naturally, if we look for solitary solutions of (1.4), we just need to get the homoclinic solutions of (1.8). Obviously, (1.8) is a special case of (1.1) with 𝑎_𝑖𝑛≡−1(𝑖=1,2,…,𝑚), 𝑏_𝑛≡2𝑚.

DNLS equation is one of the most important inherently discrete models, which models many phenomena in various areas of applications (see [2–4] and reference therein). For example, in nonlinear optics, DNLS equation appears as a model of infinite wave guide arrays. In the past decade, the existence and properties of mobile discrete solitons/breathers in DNLS equations have been considered in a number of studies [5–9].

When 𝑚=1, 𝑣_𝑛≡0, and {𝑎_𝑛}, {𝑏_𝑛}, and 𝑓(𝑛,𝑢) are 𝑇-periodic in 𝑛, the existence of homoclinic solutions for the (1.1) have been studied in [5, 6, 10] for the case where 𝑓 is with superlinear nonlinearity (kerr or cubic), in [9, 11–14] for the case where 𝑓 is with saturable nonlinearity, respectively. When {𝑎_𝑛},{𝑏_𝑛}, and 𝑓(𝑛,𝑢) are not periodic in 𝑛, the existence of homoclinic solutions for some special case of (1.1) can be found in [7, 8, 15, 16]. Especially, in [17, 18], the authors obtained sufficient conditions for the existence of at least a pair of nontrivial homoclinic solutions for the special case of (1.1) when {𝑣_𝑛} is unbounded by Nehari manifold method. It is worth pointing out that the so-called global Ambrosetti-Rabinowitz condition of 𝑓 plays a crucial role in [17, 18]. One aim of this paper is to replace the global Ambrosetti-Rabinowitz condition by a general one. The other aim of this paper is to obtain sufficient conditions for the existence of infinitely many nontrivial homoclinic solutions of (1.1). We will see that in Section 2, our results greatly improves those in [17, 18]. Our proofs of the main results are based on Mountain Pass Lemma and Fountain theorem. Our main ideas come from the papers [19–22].

This paper is organized as follows: in Section 2, we will first define some basic spaces. Then, we give the main results of this paper, and a comparison with the existing results is stated. Third, we establish the variational framework associated with (1.1) and transfer the problem of the existence and multiplicity of solutions in 𝐸 (defined in Section 2) of (1.1) into that of the existence and multiplicity of critical points of the corresponding functional. We also recall some basic results from critical point theory. Last, in Section 3, we present the proofs of our main results.

2. Preliminaries and Main Results

Let 𝑙^𝑝≡𝑙^𝑝ℤ^𝑚=𝑢=𝑢_{𝑛𝑛∈ℤ𝑚}∶∀𝑛∈ℤ^𝑚,𝑢_𝑛∈ℝ,‖𝑢‖_𝑙^𝑝=_{𝑛∈ℤ𝑚}𝑢_𝑛^𝑝1/𝑝<∞. (2.1) Then the following embedding between 𝑙^𝑝 spaces holds: 𝑙^𝑞⊂𝑙^𝑝,‖𝑢‖_𝑙^𝑝≤‖𝑢‖_𝑙^𝑞,1≤𝑞≤𝑝≤∞. (2.2)

Assume the following condition on {𝑣_𝑛} holds.

(𝑉₁) the discrete potential 𝑉={𝑣_𝑛}_𝑛∈ℤ^𝑚 satisfies lim_|𝑛|→∞𝑣_𝑛=∞. (2.3)

Let 𝐻=𝐿+𝑉. (2.4) Since the operator 𝐿 is bounded and self-adjoint in the space 𝑙²(ℤ^𝑚) with the norm ‖𝐿‖ (see [1]), and by the condition (𝑉₁), we know that the potential 𝑉 is bounded below, without loss of generality, we suppose 𝑣_𝑛>‖𝐿‖ for all 𝑛∈ℤ^𝑚. Then the operator 𝐻 is an unbounded positive self-adjoint operator in 𝑙²(ℤ^𝑚).

Define the space 𝐸∶=𝑢∈𝑙²ℤ^𝑚∶𝐻^1/2𝑢∈𝑙²ℤ^𝑚. (2.5) Then 𝐸 is a Hilbert space equipped with the norm ‖𝑢‖=𝐻^1/2𝑢_{𝑙2(𝑍𝑚)}. (2.6)

Since (𝑉₁) holds, we see that the spectrum 𝜎(𝐻) is discrete and let 𝜆₁ be the smallest eigenvalue of 𝐻, that is 𝜆₁=inf𝜎(𝐻). (2.7)

Now, we present the following basic hypotheses in order to establish the main results in this paper:

, and there exists 𝑎>0,𝑝∈(2,∞) such that 𝑓(𝑛,𝑢)≤𝑎1+|𝑢|^𝑝−1,∀𝑛∈ℤ^𝑚,𝑢∈ℝ. (2.8)

(𝑓₂)lim_|𝑢|→0𝑓(𝑛,𝑢)/𝑢=0 uniformly for 𝑛∈ℤ^𝑚.

(𝑓₃)lim_|𝑢|→∞𝐹(𝑛,𝑢)/𝑢²=+∞ uniformly for 𝑛∈ℤ^𝑚, where 𝐹(𝑛,𝑢) is the primitive function of 𝑓(𝑛,𝑢), that is, 𝐹(𝑛,𝑢)=^𝑢₀𝑓(𝑛,𝑡)𝑑𝑡. (2.9)

(𝑓₄)𝑓(𝑛,𝑢)/𝑢 is increasing in 𝑢>0 and decreasing in 𝑢<0, for all 𝑛∈ℤ^𝑚.

Under the above hypotheses, our results can be stated as follows.

Theorem 2.1. Assume that conditions (𝑉₁), (𝑓₁)–(𝑓₄) hold. Then, we have the following conclusions.(1)If 𝜎=−1, 𝜔≤𝜆₁, (1.1) has no nontrivial solution in 𝐸.(2)If 𝜎=1, 𝜔<𝜆₁, (1.1) has at least one nontrivial solution 𝑢 in 𝐸.(3)The solutions obtained in case (2) exponentially decay at infinity, that is, there exist two positive constants 𝐶 and 𝛼 such that 𝑢_𝑛≤𝐶𝑒^{−𝛼|𝑛|},𝑛∈ℤ^𝑚. (2.10)

Theorem 2.2. Assume 𝑓(𝑛,𝑢) is odd in 𝑢 for each 𝑛∈ℤ^𝑚, and that conditions (𝑉₁),(𝑓₁)–(𝑓₄) hold. Then we have the following conclusions.(1)If 𝜎=−1, 𝜔≤𝜆₁, (1.1) has no nontrivial solution in 𝐸.(2)If 𝜎=1, 𝜔<𝜆₁, (1.1) has infinitely many solutions {𝑢^(𝑘)}^∞_𝑘=1 in 𝐸 satisfying 12𝐻𝑢^(𝑘),𝑢^(𝑘)−12𝜔𝑢^(𝑘),𝑢^(𝑘)−_{𝑛∈𝑍𝑚}𝐹𝑛,𝑢_𝑛^(𝑘)→∞,𝑎𝑠𝑘→∞. (2.11)(3)The solutions obtained in case (2) exponentially decay at infinity, that is, (2.10) holds.

We notice that, in [17, 18], the authors consider the following DNLS equation 𝐻𝑢_𝑛−𝜔𝑢_𝑛−𝜎𝛾_𝑛𝑓𝑢_𝑛=0, (2.12) which is a special case of (1.1), where 𝐻=−Δ+𝑉. They obtain the following results.

Theorem A. Assume that the DNLS (2.12) satisfies (𝑉₁) and
(𝐴₁) there exist two positive constants 𝛾 and 𝛾, such that for any 𝑛∈ℤ^𝑚, 𝛾≤𝛾_𝑛≤𝛾. (2.13)
The nonlinearity 𝑓 is odd and satisfies
(𝐴₂) there are two positive constants 𝐶₁, 𝐶₂, and 2<𝑝<∞ such that 𝑓(𝑢)≤𝐶₁1+|𝑢|^𝑝−1𝑓(𝑢)(𝑣,−𝑓)≤𝐶₂1+|𝑢|^𝑝−2+|𝑣|^𝑝−2|𝑢−𝑣|. (2.14)(2.15)
(𝐴₃) lim_𝑢→0(𝑓(𝑢)/|𝑢|)=0.
(𝐴₄) there is a 2<𝑞<∞ such that 0<𝑞−1𝑓(𝑢)𝑢≤𝑓^′(𝑢)𝑢²,∀𝑢≠0. (2.16)
Then we have the following conclusions.(1)If 𝜎=−1, 𝜔≤𝜆₁, (2.12) has no nontrivial solution.(2)If 𝜎=1, 𝜔<𝜆₁, (2.12) has at least a pair of nontrivial solutions ±𝑢 in 𝑙²(ℤ^𝑚).(3)The solutions obtained in case (2) exponentially decay at infinity, that is, (2.10) holds.

Remark 2.3. Clearly, (2.12) corresponds (1.1) if we let 𝐿=−Δ,𝑓(𝑛,𝑢)=𝛾_𝑛𝑓(𝑢). (2.17) Equations (2.13) and (2.14) imply (𝑓₁), conditions (𝐴₁) and (𝐴₃) imply (𝑓₂), and conditions (𝐴₁), (𝐴₄) imply (𝑓₃) and (𝑓₄). Equation (2.15) is unnecessary in Theorem 2.2. Thus, our Theorem 2.2 greatly improves Theorem A.

Remark 2.4. In (2.12), we define 𝑓 by 𝑓(𝑢)=𝑢0,𝑢=0,|𝑢||𝑢|𝑢(|𝑢,|𝑢|1−ln,0<≤1,1+ln|)>1, (2.18) then 𝑓 does not satisfy (𝐴₄). However, if we let 𝑓(𝑛,𝑢)=𝛾_𝑛𝑓(𝑢) in (1.1), where {𝛾_𝑛} satisfies (𝐴₁), then 𝑓(𝑛,𝑢) satisfies all conditions in Theorem 2.2.

Now, we will make some preparations for the proofs of our main results. Since the operator 𝐿 is bounded in 𝑙²(ℤ^𝑚), the following two norms are equivalent in the Hilbert space 𝐸‖𝑢‖∼𝑉^1/2𝑢_{𝑙2(ℤ𝑚)}. (2.19) The following theorem plays an important role in this paper, which gives a discrete version of compact embedding theorem [16–18].

Lemma 2.5. If V satisfies the condition (𝑉₁), then for any 2≤𝑝≤∞, the embedding map from E into 𝑙^𝑝(ℤ^𝑚) is compact, denote the best embedding constant 𝑐_𝑝=max_{‖𝑢‖𝑙𝑝=1}1/‖𝑢‖.

Consider the function 𝐽 defined on 𝐸 by 𝐽(𝑢)=12((𝐻−𝜔)𝑢,𝑢)−𝜎_{𝑛∈𝑍𝑚}𝐹𝑛,𝑢_𝑛. (2.20) Standard arguments show that the functional 𝐽 is well-defined 𝐶¹ functional on 𝐸 and (1.1) is easily recognized as the corresponding Euler-Lagrange equation for 𝐽. Thus, to find nontrivial solutions of (1.1), we need only to look for nonzero critical points of 𝐽.

For the derivative of 𝐽 we have the following formula: 𝐽^′(𝑢),𝑣=((𝐻−𝜔)𝑢,𝑣)−𝜎_{𝑛∈𝑍𝑚}𝑓𝑛,𝑢_𝑛𝑣_𝑛,∀𝑣∈𝐸. (2.21)

Definition 2.6 (see [22, 23]). Let 𝐸 be a real Banach Space and 𝐽∈𝐶¹(𝐸,ℝ). For some 𝑐∈ℝ, we say 𝐽 satisfies the so-called (𝐶)_𝑐 condition if any sequence {𝑢_𝑛}⊂𝐸 such that 𝐽(𝑢_𝑛)→𝑐 and ‖𝐽^′(𝑢_𝑛)‖(1+‖𝑢_𝑛‖)→0as𝑛→∞, has a convergent subsequence.
Let 𝐵_𝑟 be the open ball in 𝐻 with radius 𝑟 and center 0, and let 𝜕𝐵_𝑟 denote its boundary. In order to obtain the existence of critical points of 𝐽 on 𝐸, we cite some basic lemmas from [24], which will be used in the proof of Theorem 2.1. The first is the following Mountain Pass Lemma.

Lemma 2.7. Let 𝐸 be a real Banach Space, 𝐽∈𝐶¹(𝐸,ℝ) satisfies the (𝐶)_𝑐 condition for any 𝑐>0,𝐽(0)=0, and
(𝐽₁) There exist 𝜌,𝜎>0 such that 𝐽∣_𝜕𝐵𝜌≥𝛼.
(𝐽₂) There exist 𝑒∈𝐸⧵𝐵_𝜌 such that 𝐽(𝑒)≤0.
Then J has a critical value 𝑐≥𝛼.

In order to prove Theorem 2.2, we shall use the following fountain theorem [23, 25, 26]. Let 𝐸 be a real Banach Space with the norm ‖⋅‖ and 𝐸=⨁_𝑗∈ℕ𝑋_𝑗 with dim 𝑋_𝑗<∞ for any 𝑗∈ℕ. Set 𝑌_𝑘=⨁^𝑘_𝑗=0𝑋_𝑗 and 𝑍_𝑘=⨁^∞_𝑗=𝑘𝑋_𝑗.

Lemma 2.8. Let 𝐽∈𝐶¹(𝐸,ℝ) be even. If, for each sufficiently large 𝑘∈ℕ, there exists 𝜌_𝑘>𝛾_𝑘>0 such that
(𝐵₁)  𝑎_𝑘∶=max_{𝑢∈𝑌𝑘,‖𝑢‖=𝜌𝑘}𝐽(𝑢)≤0.
(𝐵₂)𝑏_𝑘∶=inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝛾𝑘}𝐽(𝑢)→∞,𝑘→∞.
(𝐵₃) J satisfies the (𝐶)_𝑐 condition for every 𝑐>0.
Then J has an unbounded sequence of critical values.

3. Proofs of Main Results

Lemma 3.1. Suppose that 𝜎=1,𝜔<𝜆₁, (𝑉₁) and (𝑓₁)–(𝑓₄) hold, then we have
(i) there exists 𝜌,𝛼>0 such that 𝐽|_𝜕𝐵𝜌≥𝛼;
(ii) there exists 𝑒∈𝐸 such that 𝐽(𝑡𝑒)→−∞𝑎𝑠|𝑡|→∞.

Proof. (i) Let 𝜖=(𝜆₁−𝜔)/2. According to (𝑓₁) and (𝑓₂), it is easy to show that, there exists 𝑐₁>0, such that, 𝑓(𝑛,𝑢)≤𝜖|𝑢|+𝑐₁|𝑢|^𝑝−1 (3.1) for all 𝑛∈ℤ^𝑚 and 𝑢∈ℝ. This, together with the mean value theorem, leads to |𝐹(𝑛,𝑢)|=|𝐹(𝑛,𝑢)−𝐹(𝑛,0)|=¹₀𝑓()≤𝜖𝑛,𝑠𝑢𝑢𝑑𝑠2|𝑢|²+𝑐₁𝑝|𝑢|^𝑝. (3.2) By (3.2) and the Hölder inquality, it follows that 𝐽(𝑢)=12((𝐻−𝜔)𝑢,𝑢)−_{𝑛∈ℤ𝑚}𝐹𝑛,𝑢_𝑛≥12𝜆₁−𝜔‖𝑢‖²₂−𝜖2_{𝑛∈ℤ𝑚}𝑢_𝑛²+𝑐₁𝑝_{𝑛∈ℤ𝑚}𝑢_𝑛^𝑝≥14𝜆₁−𝜔‖𝑢‖²₂−𝑐₁𝑝‖𝑢‖^𝑝₂. (3.3) Noting that 𝑝>2, we obtain the following estimate: 𝐽(𝑢)≥18𝜆₁−𝜔𝜌²≡𝛼>0,∀‖𝑢‖=𝜌, (3.4) with 𝜌=[(𝑝/8𝑐₁)(𝜆₁−𝜔)]^1/(𝑝−2).
(ii) It follows from (𝑓₃) that for any 𝑀>0, there exists 𝛿=𝛿(𝑀)>0 such that for all 𝑛∈ℝ^𝑚,|𝑢|≥𝛿, we have 𝐹(𝑛,𝑢)≥𝑀|𝑢|². (3.5) Notice that, from (𝑓₂) and (𝑓₄), it is easy to get that 𝐹(𝑛,𝑢)>0,∀𝑢≠0. (3.6)
Let 𝑒∈𝐸 be the eigenvector of 𝐻 corresponding to the smallest eigenvalue 𝜆₁, that is to say 𝐻𝑒=𝜆₁𝑒. Then, there exists 𝑁>0, such that _{|𝑛|≤𝑁}𝑒²_𝑛≥12‖𝑒‖²₂. (3.7) Let 𝐴^∗=𝑛∈ℤ^𝑚∶|𝑛|≤𝑁,𝑒_𝑛≠0. (3.8) Taking 𝑡 large enough, such that |𝑡𝑒_𝑛|>𝛿 for all 𝑛∈𝐴^∗, then, in view of (3.5)–(3.7), we have 𝐽()=1𝑡𝑒2𝜆₁−𝜔𝑡²‖𝑒‖²₂−_{𝑛∈ℤ𝑚}𝐹𝑛,𝑡𝑒_𝑛≤12𝜆₁−𝜔𝑡²‖𝑒‖²₂−_{𝑛∈𝐴∗}𝐹𝑛,𝑡𝑒_𝑛≤12𝜆₁−𝜔𝑡²‖𝑒‖²₂−𝑀𝑡²_{𝑛∈𝐴∗}𝑒²_𝑛≤12𝜆₁−𝜔−𝑀𝑡²‖𝑒‖²₂. (3.9) Taking 𝑀 sufficiently large, for example, 𝑀≥2(𝜆₁−𝜔), we see that 𝐽(𝑡𝑒)→−∞ as |𝑡|→∞. The proof is completed.

Lemma 3.2. Suppose that 𝜎=1, 𝜔<𝜆₁, (𝑉₁),(𝑓₁)–(𝑓₄) hold. Then the functional 𝐽 satisfies the (𝐶)_𝑐 condition for any 𝑐∈ℝ.

Proof. Let {𝑢^(𝑘)}⊂𝐸 be a (𝐶)_𝑐 sequence of 𝐽, that is, 𝐽𝑢^(𝑘)→𝑐,𝐽^′𝑢^(𝑘)1+𝑢^(𝑘)→0,as𝑘→∞. (3.10) To prove the functional 𝐽 satisfies the (𝐶)_𝑐 condition, first, we prove that {𝑢^(𝑘)} is bounded in 𝐸. In fact, if not, we may assume by contradiction that ‖𝑢^(𝑘)‖→∞ as 𝑘→∞. Set 𝛼^(𝑘)∶=𝑢^(𝑘)/‖𝑢^(𝑘)‖. Up to a sequence, we have 𝛼^(𝑘)𝛼⇀𝛼in𝐸,_(𝑘)→𝛼,in𝑙^𝑞ℤ^𝑚,for𝑞≥2. (3.11)(3.12)Case  1 (𝛼≠0). By 𝐽(𝑢^(𝑘))=𝑐+𝑜(1), where 𝑜(1)→0 as 𝑘→∞, we have 𝑐+𝑜(1)=𝐽𝑢^(𝑘)=12𝑢^(𝑘)2−12𝜔𝑢^(𝑘)2₂−_{𝑛∈ℤ𝑚}𝐹𝑛,𝑢_𝑛^(𝑘). (3.13) Noticing that ‖𝑢‖²=(𝐻𝑢,𝑢)≥𝜆₁‖𝑢‖²₂, we divide both sides of (3.13) by ‖𝑢^(𝑘)‖² and get _{𝑛∈ℤ𝑚}𝐹𝑛,𝑢_𝑛^(𝑘)𝑢^(𝑘)2≤12+|𝜔|2𝜆₁−𝑐+𝑜(1)𝑢^(𝑘)2<+∞. (3.14) Let Ω={𝑛∈ℤ^𝑚∶𝛼(𝑛)≠0}, then it follows from (3.12) that 𝑢_𝑛^(𝑘)=𝛼_𝑛^(𝑘)𝑢^(𝑘)→+∞,as𝑘→∞,for𝑛∈Ω. (3.15) In view of (𝑓₃), we have lim_𝑘→∞𝐹𝑛,𝑢_𝑛^(𝑘)𝑢^(𝑘)2=lim_𝑘→∞𝐹𝑛,𝑢_𝑛^(𝑘)𝑢_𝑛^(𝑘)2𝛼_𝑛^(𝑘)2→+∞,for𝑛∈Ω. (3.16)
Therefore, _{𝑛∈ℤ𝑚}𝐹𝑛,𝑢_𝑛^(𝑘)𝑢^(𝑘)2=_𝑛∈Ω+_𝑛∉Ω𝐹𝑛,𝑢_𝑛^(𝑘)𝑢^(𝑘)2≥_𝑛∈Ω𝐹𝑛,𝑢_𝑛^(𝑘)𝑢^(𝑘)2→+∞. (3.17) This contradicts (3.14).
Case  2 (𝛼=0). We define 𝐽𝑡_𝑘𝑢^(𝑘)∶=max_[]𝑡∈0,1𝐽𝑡𝑢^(𝑘). (3.18) For any 𝑀>4, let 𝑘 be large enough such that ‖𝑢^(𝑘)‖>𝑀 and 𝛼^(𝑘)∶=2𝑀^1/2𝑢^(𝑘)/‖𝑢^(𝑘)‖=2𝑀^1/2𝛼^(𝑘).
By (3.2), (3.12), and 𝛼=0, it is clear that _{𝑛∈ℤ𝑚}𝐹𝑛,𝛼^𝑘_𝑛≤𝜖2‖𝛼‖²₂+𝑐₁𝑝‖𝛼‖^𝑝₂→0,as𝑘→∞. (3.19) Thus, for 𝑘 large enough 𝐽𝑡_𝑘𝑢^(𝑘)≥𝐽𝛼_𝑛^(𝑘)=12𝛼^(𝑘)2−12𝜔𝛼^(𝑘)2₂−_{𝑛∈ℤ𝑚}𝐹𝑛,𝛼_𝑛^(𝑘)≥12𝛼^(𝑘)2−12𝜔₀𝛼^(𝑘)2₂−_{𝑛∈ℤ𝑚}𝐹𝑛,𝛼_𝑛^(𝑘)≥12𝜔1−⁰𝜆₁𝛼^(𝑘)2−_{𝑛∈ℤ𝑚}𝐹𝑛,𝛼_𝑛^(𝑘)≥𝜔1−⁰𝜆₁𝑀, (3.20) where 𝜔₀=max{𝜔,0}<𝜆₁.
This implies that lim_𝑘→∞𝐽(𝑡_𝑘𝑢^(𝑘))=∞. Since 𝐽(0)=0 and 𝐽(𝑢^(𝑘))→𝑐as  𝑘→∞, 𝐽(𝑡𝑢^(𝑘)) attains its maximum at 𝑡_𝑘∈(0,1) for large 𝑘. Thus, (𝐽^′(𝑡_𝑘𝑢^(𝑘)),𝑢^(𝑘))=0.
On the other hand, from (𝑓₄), we have that 𝐺(𝑛,𝑠)≤𝐺(𝑛,𝑡),∀0≤𝑠<𝑡or𝑡<𝑠≤0,𝑛∈ℤ^𝑚, (3.21) where 𝐺(𝑛,𝑡)=𝑓(𝑛,𝑡)𝑡/2−𝐹(𝑛,𝑡). In fact, for 0<𝑠<𝑡  or𝑡<𝑠<0, we have 𝐺()()=()𝑛,𝑡−𝐺𝑛,𝑠𝑡𝑓𝑛,𝑡2−𝑠𝑓(𝑛,𝑠)2−^𝑡_𝑠𝑓()≥()𝑛,𝜏𝑑𝜏𝑡𝑓𝑛,𝑡2−𝑠𝑓(𝑛,𝑠)2−𝑓(𝑛,𝑡)𝑡^𝑡_𝑠≥𝑠𝜏𝑑𝜏₂2𝑓(𝑛,𝑡)𝑡−𝑓(𝑛,𝑠)𝑠>0. (3.22) If 𝑠=0, (3.21) is obvious.
By (3.10) and (3.21), we have 𝐽𝑡_𝑘𝑢^(𝑘)−12𝐽^′𝑡_𝑘𝑢^(𝑘),𝑡_𝑘𝑢^(𝑘)=_{𝑛∈ℤ𝑚}12𝑓𝑛,𝑡_𝑘𝑢_𝑛^(𝑘)𝑡_𝑘𝑢_𝑛^(𝑘)−𝐹𝑛,𝑡_𝑘𝑢_𝑛^(𝑘)≤_{𝑛∈ℤ𝑚}12𝑓𝑛,𝑢_𝑛^(𝑘)𝑢_𝑛^(𝑘)−𝐹𝑛,𝑢_𝑛^(𝑘)=𝐽𝑢^(𝑘)−12𝐽^′𝑢^(𝑘),𝑢^(𝑘)→𝑐,as𝑘→∞. (3.23) But (3.20) implies that 𝐽𝑡_𝑘𝑢^(𝑘)−12𝐽^′𝑡_𝑘𝑢^(𝑘),𝑡_𝑘𝑢^(𝑘)=𝐽𝑡_𝑘𝑢^(𝑘)→∞. (3.24) Thus, we get a contradiction. Combining the above arguments, we know that {𝑢^(𝑘)} is bounded in 𝐸.
Second, we show that there is a convergent subsequence of {𝑢^(𝑘)}. Actually, there exists a subsequence, still denoted by the same notation, such that 𝑢^(𝑘) weakly converges to some 𝑢∈𝐸. Applying Lemma 2.5, we see that that, for any 2≤𝑞≤∞, 𝑢^(𝑘)→𝑢,in𝑙^𝑞ℤ^𝑚. (3.25) By a straightforward calculation, we have 𝑢^(𝑘)−𝑢²−𝜔𝑢^(𝑘)−𝑢²₂=𝐽^′𝑢^(𝑘)−𝐽^′(𝑢),𝑢^(𝑘)−𝑢+_{𝑛∈ℤ𝑚}𝑓𝑛,𝑢_𝑛^(𝑘)−𝑓𝑛,𝑢_𝑛𝑢_𝑛^(𝑘)−𝑢_𝑛. (3.26) Due to the weak convergence, it is clear that the first term (𝐽^′(𝑢^(𝑘))−𝐽^′(𝑢),(𝑢^(𝑘)−𝑢))→0  as  𝑘→∞. It remains to show the second term in the right hand of equality (3.26) also tends to zero as 𝑘→∞.
Indeed, according to (2.2) and Hölder inequality, we have _{𝑛∈ℤ𝑚}𝑓𝑛,𝑢_𝑛^(𝑘)−𝑓𝑛,𝑢_𝑛𝑢_𝑛^(𝑘)−𝑢_𝑛≤_{𝑛∈ℤ𝑚}𝜖𝑢_𝑛^(𝑘)+𝑢_𝑛+𝑐₁𝑢_𝑛^{(𝑘)𝑝−1}+𝑢_𝑛^𝑝−1𝑢_𝑛^(𝑘)−𝑢_𝑛≤𝜖𝑢^(𝑘)₂+‖𝑢‖₂𝑢^(𝑘)−𝑢₂+𝑐₁𝑢^(𝑘)_𝑝^𝑝−1+‖𝑢‖_𝑝^𝑝−1𝑢^(𝑘)−𝑢_𝑝. (3.27) Therefore, combining (3.25) and the boundedness of {𝑢^(𝑘)}, the above inequality implies _{𝑛∈ℤ𝑚}𝑓𝑛,𝑢_𝑛^(𝑘)−𝑓𝑛,𝑢_𝑛𝑢_𝑛^(𝑘)−𝑢_𝑛→0,as𝑘→∞. (3.28) So, from (3.26) we can conclude that {𝑢^(𝑘)}→𝑢 in 𝐸, and this means 𝐽 satisfies (𝐶)_𝑐 condition. The proof is completed.

Proof of Theorem 2.1. (1) By way of contradiction, assume that (1.1) has a nontrivial solution 𝑢∈𝐸. Then 𝑢 is a nonzero critical point of 𝐽  in  𝐸. Thus, 𝐽^′(𝑢)=0. But 𝐽^′(𝑢),𝑢=((𝐻−𝜔)𝑢,𝑢)−𝜎_{𝑛∈ℤ𝑚}𝑓𝑛,𝑢_𝑛𝑢_𝑛≥_{𝑛∈ℤ𝑚}𝑓𝑛,𝑢_𝑛𝑢_𝑛>0. (3.29) This is a contradiction, so the conclusion holds.
(2) By Lemma 3.1, the functional 𝐽 satisfies (𝐽₁) and (𝐽₂) of the mountain pass theorem. Lemma 3.2 implies that 𝐽 satisfies (𝐶)_𝑐 condition for any 𝑐∈ℝ. It follows from Lemma 2.7 that 𝐽 has a critical value 𝑐≥𝛼>0. Hence, (1.1) has at least one nontrivial solution 𝑢∈𝐸.
(3) Assume 𝑤 is a nontrivial solution obtained in (2). Similar to [17], let Θ={𝜃_𝑛≡−𝜎𝑓(𝑛,𝑤_𝑛)/𝑤_𝑛} and the operator 𝐺 defined by (𝐺𝑢)_𝑛=(𝐻𝑢)_𝑛+𝜃_𝑛𝑢_𝑛 for 𝑢∈𝐸, then (1.1) is equivalent to 𝐺𝑢=𝜔𝑢. (3.30) Since 𝑤∈𝐸, we have lim_|𝑛|→∞𝜃_𝑛=0. The multiplication operator Θ is compact in 𝐸⊂𝑙²(ℤ^𝑚), and this implies that 𝜎_ess(𝐺)=𝜎_ess(𝐻)=𝜙. (3.31) Equation (2.10) follows from the standard theorem on exponential decay for the eigenfunction of Jacobi operator (see [1] for details).
This completes the proof of Theorem 2.1.

Assume that 𝑖∈ℤ^𝑚. Define 𝑒^(𝑖)={𝑒_𝑛^(𝑖)} by 𝑒_𝑛^(𝑖)=1,𝑛=𝑖,0,𝑛≠𝑖. (3.32)

Let 𝑋_𝑗=span{𝑒^(𝑛)∶|𝑛|=𝑗} for 𝑗∈ℕ={0,1,2,…}, 𝑌_𝑘=⨁^𝑘_𝑗=0𝑋_𝑗, and 𝑍_𝑘=⨁^∞_𝑗=𝑘𝑋_𝑗, then we have

Lemma 3.3. Suppose that 𝜎=1,𝜔<𝜆₁, (𝑉₁),(𝑓₁)–(𝑓₃) hold. Then there exists 𝜌_𝑘>𝛾_𝑘>0 such that(i)𝑏_𝑘∶=inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝛾𝑘}𝐽(𝑢)→∞,as𝑘→∞, (ii)𝑎_𝑘∶=max_{𝑢∈𝑌𝑘,‖𝑢‖=𝜌𝑘}𝐽(𝑢)≤0.

Proof. (i) It follows from (3.2) that, for any 𝑢∈𝑍_𝑘𝐽(𝑢)=12()−1𝐻𝑢,𝑢2𝜔(𝑢,𝑢)−_{𝑛∈ℤ𝑚}𝐹𝑛,𝑢_𝑛≥12‖𝑢‖²−12(𝜔+𝜖)‖𝑢‖²₂−𝑐₁𝑝‖𝑢‖^𝑝_𝑝≥121−(|𝜔|+𝜖)𝛽²_𝑘‖𝑢‖²−𝑐₁𝑝𝛽^𝑝_𝑘‖𝑢‖^𝑝, (3.33) where 𝛽_𝑘=sup_{𝑢∈𝑍𝑘,‖𝑢‖=1}‖𝑢‖₂. Since for 𝑢∈𝑍_𝑘, ‖𝑢‖²=(𝐻𝑢,𝑢)=(𝐿𝑢,𝑢)+(𝑉𝑢,𝑢)≥−‖𝐿‖‖𝑢‖²₂+_{|𝑛|≥𝑘}𝑣_𝑛𝑢²_𝑛≥inf_{|𝑛|≥𝑘}𝑣_𝑛−‖𝐿‖‖𝑢‖²₂, (3.34) we see that 𝛽_𝑘→0 as 𝑘→∞. Thus, 𝑏_𝑘∶=inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝛾𝑘}𝐽(𝑢)≥inf_{𝑢∈𝑍𝑘,‖𝑢‖=𝛾𝑘}121−(|𝜔|+𝜖)𝛽²_𝑘‖𝑢‖²−𝑐₁𝑝𝛽^𝑝_𝑘‖𝑢‖^𝑝=141−(|𝜔|+𝜖)𝛽²_𝑘𝛾²_𝑘, (3.35) where 𝛾_𝑘=[𝑝(1−|𝜔|𝛽²_𝑘−𝜖𝛽²_𝑘)/4𝑐₁𝛽^𝑝_𝑘]^1/(𝑝−2). Notice that 𝑝>2 and 𝛽_𝑘→0 as 𝑘→∞, so we have 𝑏_𝑘→∞, as 𝑘→∞.
(ii) For any 𝑘∈ℕ, let the dimension of 𝑌_𝑘 be 𝜒_𝑘 and 𝑣^∗_𝑘=max_{|𝑗|≤𝑘}{𝑣_𝑗}. Assume that 𝑀=‖𝐿‖+𝑣^∗_𝑘−𝜔, then from (3.5), there exists a 𝛿_𝑘>0 such that 𝐹(𝑛,𝑢)≥𝑀|𝑢|² for |𝑢|≥𝛿_𝑘. Thus, for 𝑢∈𝑌_𝑘, 𝐽(𝑢)=12((𝐻−𝜔)𝑢,𝑢)−_{|𝑛|≤𝑘}𝐹𝑛,𝑢_𝑛=12((𝐿+𝑉−𝜔)𝑢,𝑢)−_{𝑢𝑛≤𝛿𝑘}𝐹𝑛,𝑢_𝑛−_{𝑢𝑛>𝛿𝑘}𝐹𝑛,𝑢_𝑛≤12‖𝐿‖+𝑣^∗_𝑘−𝜔‖𝑢‖²₂−𝑀_{𝑢𝑛>𝛿𝑘}𝑢²_𝑛≤12‖𝐿‖+𝑣^∗_𝑘−𝜔‖𝑢‖²₂−𝑀_{|𝑛|≤𝑘}𝑢²_𝑛+𝑀_{𝑢𝑛≤𝛿𝑘}𝑢²_𝑛≤12‖𝐿‖+𝑣^∗_𝑘−𝜔−2𝑀‖𝑢‖²₂+𝑀𝛿²_𝑘𝜒_𝑘1=−2‖𝐿‖+𝑣^∗_𝑘−𝜔‖𝑢‖²₂+‖𝐿‖+𝑣^∗_𝑘−𝜔𝛿²_𝑘𝜒_𝑘. (3.36) Taking 𝜌_𝑘 sufficiently large, we have, 𝑎_𝑘∶=max_{𝑢∈𝑌𝑘,‖𝑢‖=𝜌𝑘}𝐽(𝑢)≤0. (3.37) The proof is completed.

Proof of Theorem 2.2. The proofs for (1) and (3) are similar to that of (1) and (3) in Theorem 2.1, and we omit them. Now we give the proof of (2). By Lemma 3.3, the functional 𝐽 satisfies (𝐵₁) and (𝐵₂) of Lemma 2.8. Lemma 3.2 implies that 𝐽 satisfies (𝐶)_𝑐 condition for any 𝑐∈ℝ. 𝑓 is odd implies that 𝐽(𝑢) is even. It follows from Lemma 2.8 that 𝐽 has a sequence of critical points {𝑢^(𝑘)}⊂𝐸, such that 𝐽(𝑢^(𝑘))→∞. Hence, (1.1) has infinitely many high-energy solutions in 𝑙²(ℤ^𝑚). This completes the proof.

Acknowledgments

The authors are grateful to the anonymous referee for his/her valuable suggestions. This work is supported by the National Natural Science Foundation of China (no. 11171078) and the Specialized Fund for the Doctoral Program of Higher Education of China (no. 20114410110002).