Homoclinic Solutions of a Class of Nonperiodic Discrete Nonlinear Systems in Infinite Higher Dimensional Lattices

Abstract

By using critical point theory, we obtain a new sufficient condition on the existence of homoclinic solutions of a class of nonperiodic discrete nonlinear systems in infinite lattices. The classical Ambrosetti-Rabinowitz superlinear condition is improved by a general superlinear one. Some results in the literature are improved.

1. Introduction

Assume that 𝑚 is a positive integer. Consider the following difference equation in infinite higher dimensional lattices: 𝐿𝑢_𝑛−𝜔𝑢_𝑛=𝜎𝛾_𝑛𝑓_𝑛𝑢_𝑛,𝑛∈𝑍^𝑚, (1) where 𝑓_𝑛(𝑢) is continuous in 𝑢, 𝜎=±1, 𝑛=(𝑛₁,𝑛₂,…,𝑛_𝑚)∈𝑍^𝑚, 𝛾={𝛾_𝑛} is a positive real valued sequence, 𝜔∈𝑅, and 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)}; (2) here, {𝑎_𝑖𝑛}(𝑖=1,2,…,𝑚) and {𝑏_𝑛} are real valued bounded sequences.

Assume that 𝑓_𝑛(0)=0 for 𝑛∈𝑍^𝑚; then {𝑢_𝑛}={0} is a solution of (1), which is called the trivial solution. As usual, we say that 𝑢={𝑢_𝑛}, a solution of (1), is homoclinic (to 0) if lim_|𝑛|→∞𝑢_𝑛=0, (3) where |𝑛|=|𝑛₁|+|𝑛₂|+⋯+|𝑛_𝑚| is the length of multi-index 𝑛. In addition, if {𝑢_𝑛}≠{0}, then 𝑢 is called a nontrivial homoclinic solution. We are interested in the existence of the nontrivial homoclinic solutions for (1). This problem appears when we seek the discrete solitons of nonperiodic discrete nonlinear Schrödinger (DNLS) equation 𝑖𝜓̇_𝑛=−Δ𝜓_𝑛−𝜎𝛾_𝑛𝑓_𝑛𝜓_𝑛,𝑛∈𝑍^𝑚, (4) where 𝜎=±1 and Δ𝜓_𝑛=𝜓_{(𝑛1+1,𝑛2,…,𝑛𝑚)}+𝜓_{(𝑛1−1,𝑛2,…,𝑛𝑚)}+𝜓_{(𝑛1,𝑛2+1,…,𝑛𝑚)}+𝜓_{(𝑛1,𝑛2−1,…,𝑛𝑚)}+⋯+𝜓_{(𝑛1,𝑛2,…,𝑛𝑚+1)}+𝜓_{(𝑛1,𝑛2,…,𝑛𝑚−1)}−2𝑚𝜓_{(𝑛1,𝑛2,…,𝑛𝑚)} (5) is the discrete Laplacian in 𝑚 spatial dimension. Typical representatives of power nonlinearities are 𝑓_𝑛(𝑢)=𝑙_𝑛|𝑢|^𝑝𝑢,𝑙_𝑛,𝑝>0. (6)

Primarily, we are interested in spatially localized, or solitary, standing waves. Such waves are often called breathers or gap solitons. The origin of the last name is that typically such solutions do exist for frequencies in gaps of linear spectrum. Considering (4), we suppose that the nonlinearity is gauge invariant; that is, 𝑓_𝑛𝑒^𝑖𝜃𝑢=𝑒^𝑖𝜃𝑓_𝑛(𝑢),𝜃∈𝑅, (7) and, in addition, 𝑓_𝑛(𝑢)≥0 for 𝑢≥0 for 𝑛∈𝑍^𝑚.

Making use of the standing wave ansatz, 𝜓_𝑛=𝑢_𝑛𝑒^{−𝑖𝜔𝑡},lim_|𝑛|→∞𝜓_𝑛=0, (8) where {𝑢_𝑛} is a real valued sequence and 𝜔∈𝑅 is the temporal frequency. Then (4) becomes −Δ𝑢_𝑛−𝜔𝑢_𝑛=𝜎𝛾_𝑛𝑓_𝑛𝑢_𝑛,𝑛∈𝑍^𝑚, (9) and (3) holds. This is an equation of the form (1) with 𝑎_𝑖𝑛=−1(𝑖=1,2,…,𝑚) and 𝑏_𝑛=2𝑚.

When 𝑓_𝑛(𝑢) has the form of (6), the homoclinic solutions of (9) were obtained by Karachalios in [2] by assuming that 𝛾∈𝑙^𝜌, 𝜌=(𝑞−1)/(𝑞−2), for some 𝑞>2. We note that 𝛾∈𝑙^𝜌 implies that lim_|𝑛|→∞𝛾_𝑛=0. Moreover, (6) satisfies the classical Ambrosetti-Rabinowitz superlinear condition [3], and 𝑓_𝑛(𝑢)/|𝑢| is nondecreasing with respect to |𝑢|, both of which played important roles in the existence of homoclinic solutions of (1.7) in [2].

The aim of this paper is to improve both the monotone condition of 𝑓_𝑛(𝑢)/|𝑢| and the classical Ambrosetti-Rabinowitz superlinear condition by general ones; see Remarks 8 and 9 for details. Moreover, in this paper, we only need lim_|𝑛|→∞𝛾_𝑛=0. Particularly, our results improved the results in [2]; see Remarks 3 and 7 for details.

In the past years, there has been large growth in the study of DNLS equation, which is a nonlinear lattice system that appears in many areas of physics. Discrete solitons which exist in DNLS systems, that is, solitary waves and localized structures in spatially discrete media, are also of particular interest in their own right. Among these, one can mention photorefractive media [4], biomolecular chains [5], and Bose-Einstein condensates [6]. The experimental observations of discrete solitons in nonlinear lattice systems have been reported [7–11]. To mention that, many authors have studied the existence of discrete solitons of the DNLS equations [12–17]. The fruitful methods include centre manifold reduction [16], variational methods [12, 14], the principle of anticontinuity [13, 17], and the Nehari manifold approach [18]. However, most of the existing literature is devoted to the DNLS equations with constant coefficients or periodic coefficients. Results on such DNLS equations have been summarized in [19–23]. And we also want to mention that, in recent years, the existence of homoclinic solutions for difference equations has been studied by many authors, and we refer to [24–36].

Since the operator 𝐿 is bounded and self-adjoint in the space 𝑙² (defined in Section 2), we consider (1) as a nonlinear equation in 𝑙² with (3) being satisfied automatically. The spectrum 𝜎(𝐿) of 𝐿 is closed. Thus, the complement 𝑅⧵𝜎(𝐿) consists of a finite number of open intervals called spectral gaps and two of them are semi-infinite which are denoted by (−∞,𝛽) and (𝛼,∞), respectively. In this paper, we consider the homoclinic solutions of (1) in 𝑙² for the case where 𝜔∈(−∞,𝛽) and 𝜎=1. The case where 𝜔∈(𝛼,∞) and 𝜎=−1 is omitted, since, in this case, we can replace 𝐿 by −𝐿.

The main idea in this paper is as follows. First, we assume that {𝛾_𝑛} converges to zero at infinity; that is, lim_|𝑛|→∞𝛾_𝑛=0. After that, we prove a compact inclusion between ordinary sequence spaces 𝑙² and weighted sequence spaces 𝑙²_𝛾 (defined in Section 2), in order to come over lack of compactness for the so-called (𝐶)_𝑐 condition (defined in Section 2). Finally, by making use of the Mountain Pass Lemma [37], we prove the existence of homoclinic solutions of (1) in 𝑙².

2. Preliminaries

In this section, we first establish the variational setting associated with (1). Let 𝑙^𝑝=𝑙^𝑝𝑍^𝑚=𝑢=𝑢_{𝑛𝑛∈𝑍𝑚}∶∀𝑛∈𝑍^𝑚,𝑢_𝑛‖𝑢‖∈𝑅,_𝑙𝑝=_{𝑛∈𝑍𝑚}𝑢_𝑛^𝑝1/𝑝<∞. (10) Then the following embedding between 𝑙^𝑝 spaces holds: 𝑙^𝑞⊂𝑙^𝑝,‖𝑢‖_𝑙^𝑝≤‖𝑢‖_𝑙^𝑞,1≤𝑞≤𝑝≤∞. (11) For 𝑝=2, we get the usual Hilbert space of square-summable sequences, with the real scalar product (𝑢,𝑣)_𝑙²=_{𝑛∈𝑍𝑚}𝑢_𝑛𝑣_𝑛,𝑢,𝑣∈𝑙². (12)

For a positive real valued bounded sequence 𝛾={𝛾_𝑛∶0<𝛾_𝑛≤𝛾<∞}_{𝑛∈𝑍𝑚}, we define the weighted sequence spaces 𝑙²_𝛾: 𝑙²_𝛾=𝑢=𝑢_{𝑛𝑛∈𝑍𝑚}∶∀𝑛∈𝑍^𝑚,𝑢_𝑛‖𝑢‖∈𝑅,_𝑙2𝛾=_{𝑛∈𝑍𝑚}𝛾_𝑛𝑢_𝑛^21/2<∞. (13) It is not hard to see that 𝑙²_𝛾 is a Hilbert space, with the scalar product (𝑢,𝑣)_𝑙²_𝛾=_{𝑛∈𝑍𝑚}𝛾_𝑛𝑢_𝑛𝑣_𝑛,𝑢,𝑣∈𝑙²_𝛾. (14)

For a certain class of weight 𝛾, we have the following lemmas, which will play a crucial role in our analysis.

Lemma 1. Let 𝜅={𝜅_𝑛∶|𝜅_𝑛|≤𝜅<∞}_𝑛∈𝑍^𝑚 be a multiplication operator from 𝑙²_𝛾 to 𝑙²_𝛾 defined by 𝜅𝑢={𝜅_𝑛𝑢_𝑛}_𝑛∈𝑍^𝑚. If lim_|𝑛|→∞𝜅_𝑛=0, then the operator 𝜅 is compact.

Proof of Lemma 1. Let Λ=𝜅𝑢∶‖𝑢‖_𝑙²_𝛾≤1. (15) We only need to prove that Λ is precompact in 𝑙²_𝛾. By assumption, for any 𝜀>0, there exists 𝑁>0 such that |𝜅_𝑛|≤𝜀 for any |𝑛|>𝑁. Define a cutting sequence 𝜒={𝜒_𝑛} by 𝜒_𝑛=|𝑛||𝑛|1,≤𝑁,0,>𝑁. (16) Denote by 𝜒^𝑐=1−𝜒 the anticutting sequence. Then for any 𝜅𝑢∈Λ𝜒^𝑐𝜅𝑢²_𝑙²_𝛾=_|𝑛|>𝑁𝛾_𝑛𝜅_𝑛𝑢_𝑛²≤𝜀²,𝜒𝜅𝑢²_𝑙²_𝛾=_{|𝑛|≤𝑁}𝛾_𝑛𝜅_𝑛𝑢_𝑛²≤𝜅². (17) For arbitrary 𝜀>0 and Λ_𝜀={𝜒𝜅𝑢∶‖𝑢‖_𝑙²_𝛾≤1} finite-dimensional and bounded, we know that Λ is precompact. The proof is complete.

Lemma 2. One assumes positive sequence of real numbers 𝛾 with lim_|𝑛|→∞𝛾_𝑛=0. Then 𝑙²↪𝑙²_𝛾 with compact inclusion.

Proof of Lemma 2. Note that ‖𝑢‖_𝑙²_𝛾≤𝛾‖𝑢‖_𝑙² for any 𝑢∈𝑙² and 𝛾^1/2 is compact. Thus, 𝑙²↪𝑙²_𝛾 with compact inclusion by Lemma 1. The proof is complete.

Remark 3. Karachalios [2] proved 𝑙²↪𝑙²_𝛾 with compact inclusion assuming that 𝛾∈𝑙^𝜌, 𝜌=(𝑞−1)/(𝑞−2), for some 𝑞>2. Note that 𝛾∈𝑙^𝜌 implies that lim_|𝑛|→∞𝛾_𝑛=0. Thus, we find that Lemma 2 improves Lemma 2.1 in [2].

On the Hilbert space 𝑙², we consider the functional 𝐽(𝑢)=12((𝐿−𝜔)𝑢,𝑢)_𝑙²−𝜎_{𝑛∈𝑍𝑚}𝛾_𝑛𝐹_𝑛𝑢_𝑛, (18) where 𝐹_𝑛(𝑢)=^𝑢₀𝑓_𝑛(𝑠)𝑑𝑠 (19) is the primitive function of 𝑓_𝑛(𝑢). Then 𝐽∈𝐶¹(𝑙²,𝑅) and 𝐽^′(𝑢),𝑣=((𝐿−𝜔)𝑢,𝑣)_𝑙²−𝜎_{𝑛∈𝑍𝑚}𝛾_𝑛𝑓_𝑛𝑢_𝑛𝑣_𝑛,𝑢,𝑣∈𝑙². (20)

Equation (20) implies that (1) is the corresponding Euler-Lagrange equation for 𝐽. Therefore, we have reduced the problem of finding a nontrivial homoclinic solution of (1) to that of seeking a nonzero critical point of the functional 𝐽 on 𝑙².

Let 𝛿 be the distance from 𝜔 to the spectrum 𝜎(𝐿); that is, 𝛿=𝛽−𝜔. (21) Then, we have ((𝐿−𝜔)𝑢,𝑢)≥𝛿‖𝑢‖²_𝑙2,𝑢∈𝑙². (22) We also consider a norm in 𝑙² defined by ‖𝑢‖_𝑙²_𝜔=[((𝐿−𝜔)𝑢,𝑢)]^1/2,𝑢∈𝑙². (23) Since 𝛿‖𝑢‖²_𝑙2≤‖𝑢‖²_𝑙2𝜔≤‖𝐿−𝜔‖‖𝑢‖²_𝑙2,𝑢∈𝑙², (24) norm (23) is an equivalent norm with the usual one of 𝑙².

In order to obtain the existence of critical points of 𝐽 on 𝑙², we cite some basic notations and some known results from critical point theory.

Let 𝐻 be a Hilbert space and 𝐶¹(𝐻,𝑅) denote the set of functionals that are Fréchet differentiable and their Fréchet derivatives are continuous on 𝐻.

Let 𝐽∈𝐶¹(𝐻,𝑅). A sequence {𝑢_𝑗}⊂𝐻 is called a (𝐶)_𝑐 sequence for 𝐽 if 𝐽(𝑢_𝑗)→𝑐 for some 𝑐∈𝑅 and (1+‖𝑢_𝑗‖)‖𝐽^′(𝑢_𝑗)‖ as 𝑗→∞. We say 𝐽 satisfies the (𝐶)_𝑐 condition if any (𝐶)_𝑐 sequence for 𝐽 possesses 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. The following lemma is taken from [37].

Lemma 4 (Mountain Pass Lemma). If 𝐽∈𝐶¹(𝐻,𝑅) and satisfies the following conditions: there exist 𝑒∈𝐻⧵{0} and 𝑟∈(0,‖𝑒‖) such that max{𝐽(0),𝐽(𝑒)}<inf_{‖𝑢‖=𝑟}𝐽(𝑢), then there exists a (𝐶)_𝑐 sequence {𝑢_𝑛} for the mountain pass level 𝑐 which is defined by 𝑐=inf_ℎ∈Γmax_[]𝑠∈0,1𝐽(ℎ(𝑠)), (25) where Γ={ℎ∈𝐶([0,1],𝐻)∶ℎ(0)=0,ℎ(1)=𝑒}. (26)

3. Main Results

In this section, we will establish some sufficient conditions on the existence of nontrivial solutions of (1) in 𝑙².

Theorem 5. Assume that 𝜎=1, 𝜔∈(−∞,𝛽), and the following conditions hold. (𝐻1)𝑓_𝑛(𝑢) is continuous in 𝑢, 𝑓_𝑛(𝑢)=𝑜(𝑢) as 𝑢→0 uniformly for 𝑛∈𝑍^𝑚.(𝐻2)There exist 𝑏>0, 𝑝>2 such that 𝑓_𝑛(𝑢)≤𝑏1+|𝑢|^𝑝−1 (27) uniformly for 𝑛∈𝑍^𝑚 and 𝑢∈𝑅.(𝐻3)There exists some 𝜃≥1 such that 𝜃𝐺_𝑛(𝑢)≥𝐺_𝑛(𝑡𝑢), for 𝑛∈𝑍^𝑚, 𝑢∈𝑅, and 𝑡∈[0,1], where 𝐺_𝑛(𝑢)=(1/2)𝑓_𝑛(𝑢)𝑢−𝐹_𝑛(𝑢).(𝐻4)𝐹_𝑛(𝑢)≥0 for 𝑢∈𝑅, and lim_|𝑢|→∞(𝐹_𝑛(𝑢)/𝑢²)=∞, uniformly for 𝑛∈𝑍^𝑚.(𝐻5)Positive real valued sequence 𝛾={𝛾_𝑛}_𝑛∈𝑍^𝑚 with lim_|𝑛|→∞𝛾_𝑛=0.Then (1) has at least a nontrivial solution 𝑢 in 𝑙² and the solution decays exponentially at infinity. That is, there exist two positive constants 𝐶 and 𝜏 such that 𝑢_𝑛≤𝐶𝑒^{−𝜏|𝑛|},𝑛∈𝑍^𝑚. (28)

Theorem 5 gives some sufficient conditions on the existence of nontrivial solutions of (1) in 𝑙². However, (1) may have no nontrivial solutions in 𝑙². In fact, we have the following proposition.

Proposition 6. Assume that 𝜎=−1, 𝜔≤𝛽, and 𝛾_𝑛𝑓_𝑛(𝑢)𝑢>0 when 𝑢≠0 for all 𝑛∈𝑍^𝑚. Then (1) has no nontrivial solutions in 𝑙².

Proof of Proposition 6. By way of contradiction, we assume that (1) has a nontrivial solution 𝑢={𝑢_𝑛}∈𝑙². Then 𝑢 is a nonzero critical point of 𝐽, and 𝐽^′(𝑢),𝑢=((𝐿−𝜔)𝑢,𝑢)_𝑙²−𝜎_{𝑛∈𝑍𝑚}𝛾_𝑛𝑓_𝑛𝑢_𝑛𝑢_𝑛≥_{𝑛∈𝑍𝑚}𝛾_𝑛𝑓_𝑛𝑢_𝑛𝑢_𝑛>0. (29) This is a contradiction as ⟨𝐽^′(𝑢),𝑢⟩=0, so the conclusion holds.

Remark 7. It is easy to see that the function 𝑓_𝑛 defined by 𝑓_𝑛(𝑢)=|𝑢|^2𝜆𝑢, (30) where 𝜆>0 and 𝛾∈𝑙^𝜌, 𝜌=(𝑞−1)/(𝑞−2), for some 𝑞>2, satisfies all conditions in Theorem 5. This case was studied by [2], and we find that Theorem 5 improves Theorem 2.3 in [2].

Remark 8. We will introduce another condition (𝐻3): 𝑓_𝑛(𝑢)/|𝑢| is nondecreasing with respect to |𝑢|. We want to point out that condition (𝐻3) is equivalent to (𝐻3) when 𝜃=1 and (𝐻3) gives “better monotony” when 𝜃>1, since (𝐻3) implies (𝐻3) (see [38]). Moreover, we can find that 𝑓_𝑛(𝑢)=5𝑢ln(1+𝑢²)+9sin𝑢 satisfies (𝐻3) but not (𝐻3) for some 𝜃≥100.

Remark 9. As we know, the condition (𝐻4)𝑓(𝑢)𝑢>𝑞𝐹(𝑢)>0forsome𝑞>2and𝑢≠0is often called Ambrosetti-Rabinowitz superlinear condition [3]. Clearly, (𝐻4) implies (𝐻4). Actually, (𝐻4) is more general than (𝐻4). Let {𝑎_𝑛} be a positive sequence, and 𝑓_𝑛(𝑢)=𝑎_𝑛𝑢ln(1+|𝑢|). Then 𝑓_𝑛 satisfies (𝐻4). However, 𝑓_𝑛 does not satisfy (𝐻4).

The proof of Theorem 5 is based on a direct application of the following lemmas. The key points read as follows.

Lemma 10. Assume that the conditions of Theorem 5 hold; then one has the following. (𝐽₁)There exist two constants 𝑎>0 and 𝜌>0 such that 𝐽|_𝜕𝐵𝜌≥𝑎.(𝐽₂)There exists an 𝑒∈𝑙² such that 𝐽(𝑡𝑒)→−∞ as |𝑡|→∞.

Proof of Lemma 10. Let 𝛾=max_𝑛∈𝑍^𝑚{𝛾_𝑛} and 𝜖=𝛿/2𝛾. By (𝐻1) and (𝐻2), there exists 𝑐₁>0, such that 𝑓_𝑛(𝑢)≤𝜖|𝑢|+𝑐₁|𝑢|^𝑝−1 (31) for all 𝑛∈𝑍^𝑚 and 𝑢∈𝑅, and (31) implies that 𝐹_𝑛(𝑢≤𝜖)2|𝑢|²+𝑐₁𝑝|𝑢|^𝑝. (32) By (32) and the Hölder inequality, we have 𝐽(𝑢)=12((𝐿−𝜔)𝑢,𝑢)_𝑙²−_{𝑛∈𝑍𝑚}𝛾_𝑛𝐹_𝑛𝑢_𝑛≥𝛿2‖𝑢‖²_𝑙2−𝛾_{𝑛∈𝑍𝑚}𝜖2|𝑢|²𝑐₁𝑝|𝑢|^𝑝≥𝛿4‖𝑢‖²_𝑙2−𝑐₁𝛾𝑝‖𝑢‖^𝑝_𝑙2. (33) Since 𝑝>2, we have 𝐽(𝑢)≥𝛿𝜌²8=𝑎>0for‖𝑢‖_𝑙²=𝜌, (34) where 𝜌=(𝛿𝑝/8𝑐₁𝛾)^1/(𝑝−2).
Let 𝑒={𝑒_𝑛}∈𝑙² be the eigenvector of 𝐿 corresponding to the eigenvalue 𝛽; that is to say, 𝐿𝑒=𝛽𝑒. There exists 𝑁>0, such that _{|𝑛|≤𝑁}𝑒²_𝑛≥12‖𝑒‖²_𝑙2. (35) Let 𝐴^∗=𝑛∈𝑍^𝑚∶|𝑛|≤𝑁,𝑒_𝑛≠0. (36) By (𝐻4), for any 𝑀>0, there exists 𝜂=𝜂(𝑀)>0 such that 𝐹_𝑛(𝑢)≥𝑀|𝑢|²for𝑛∈𝑍^𝑚,|𝑢|≥𝜂. (37) Taking 𝑡 large enough, such that 𝑡𝑒_𝑛>𝜂 for all 𝑛∈𝐴^∗, then, combining (35), (36), and (37), we have 𝐽()=1𝑡𝑒2((𝐿−𝜔)𝑡𝑒,𝑡𝑒)_𝑙²−_{𝑛∈𝑍𝑚}𝛾_𝑛𝐹_𝑛𝑡𝑒_𝑛≤𝛿2𝑡²‖𝑒‖²_𝑙2−_{𝑛∈𝐴∗}𝛾_𝑛𝐹_𝑛𝑡𝑒_𝑛≤𝛿2𝑡²‖𝑒‖²_𝑙2−𝛾𝑀𝑡²_{𝑛∈𝐴∗}𝑒²_𝑛≤12𝛿−𝛾𝑀𝑡²‖𝑒‖²_𝑙2, (38) where 𝛾=min_𝑛∈𝐴^∗{𝛾_𝑛}>0. Letting 𝑀 be large enough, such that 𝛿≤𝛾𝑀, we obtain that 𝐽(𝑡𝑒)→−∞ as |𝑡|→∞. The proof is complete.

Lemma 11. Assume that the conditions of Theorem 5 hold; then the functional 𝐽 satisfies the (𝐶)_𝑐 condition for any given 𝑐∈𝑅.

Proof of Lemma 11. Let {𝑢^(𝑘)}⊂𝑙² be a (𝐶)_𝑐 sequence of 𝐽; that is, 𝐽𝑢^(𝑘)𝑢⟶𝑐,1+^(𝑘)_𝑙2𝐽^′𝑢^(𝑘)_𝑙2⟶0,as𝑘⟶∞. (39) First, we prove that {𝑢^(𝑘)} is bounded in 𝑙². By way of contradiction, assume that ‖𝑢^(𝑘)‖_𝑙²→∞ as 𝑘→∞. Set 𝜉^(𝑘)=𝑢^(𝑘)/‖𝑢^(𝑘)‖_𝑙². Up to a sequence, we have 𝜉^(𝑘)⇀𝜉,in𝑙²,𝜉^(𝑘)⟶𝜉,in𝑙²_𝛾. (40)(41)
Case 1   (𝜉≠0). By 𝐽(𝑢^(𝑘))=𝑐+𝑜(1), where 𝑜(1)→0 as 𝑘→0, we have _{𝑛∈𝑍𝑚}𝛾_𝑛𝐹_𝑛𝑢_𝑛^(𝑘)𝑢^(𝑘)2_𝑙²=12(𝐿−𝜔)𝑢^(𝑘),𝑢^(𝑘)𝑢^(𝑘)2_𝑙²−𝑐+𝑜(1)𝑢^(𝑘)2_𝑙²≤‖𝐿−𝜔‖2−𝑐+𝑜(1)𝑢^(𝑘)2_𝑙²<∞. (42) Let 𝐵^∗={𝑛∈𝑍^𝑚∶𝜉_𝑛≠0}. Obviously, 𝐵^∗ is nonempty. Then, for some 𝑛₀∈𝐵^∗, it follows from (41) that 𝑢_𝑛^(𝑘)₀=𝜉_𝑛^(𝑘)₀𝑢^(𝑘)_𝑙2⟶∞,as𝑘⟶∞. (43) Combining (𝐻4) and 𝛾_𝑛0>0, we have 𝛾_𝑛0𝐹_𝑛0𝑢_𝑛^(𝑘)₀𝑢^(𝑘)2_𝑙²=𝛾_𝑛0𝐹_𝑛0𝑢_𝑛^(𝑘)₀𝑢_𝑛^(𝑘)₀²𝜉_𝑛^(𝑘)₀²⟶∞,as𝑘⟶∞. (44) However, _{𝑛∈𝑍𝑚}𝛾_𝑛𝐹_𝑛𝑢_𝑛^(𝑘)𝑢^(𝑘)2_𝑙²=_𝑛≠𝑛0𝛾_𝑛𝐹_𝑛𝑢_𝑛^(𝑘)𝑢^(𝑘)2_𝑙²+𝛾_𝑛0𝐹_𝑛0𝑢_𝑛^(𝑘)₀𝑢^(𝑘)2_𝑙²≥𝛾_𝑛0𝐹_𝑛0𝑢_𝑛^(𝑘)₀𝑢^(𝑘)2_𝑙²⟶∞, (45) as 𝑘→∞. This contradicts (42).
Case 2  (𝜉=0). Let 𝐽𝑡_𝑘𝑢^(𝑘)=max_[]𝑡∈0,1𝐽𝑡𝑢^(𝑘). (46) For any given 𝑀>max{4,𝜃𝑐/2𝛿}, let 𝑘 be large enough such that ‖𝑢^(𝑘)‖_𝑙²>𝑀 and 𝜉^(𝑘)=2𝑀^1/2𝜉^(𝑘). Combining (32), (41), and 𝜉=0, it is easy to see that _{𝑛∈𝑍𝑚}𝛾_𝑛𝐹_𝑛𝜉_𝑛^(𝑘)≤𝜖2𝜉^(𝑘)2_𝑙2𝛾+𝑐₁𝛾^1/2𝑝𝜉^(𝑘)_𝑙^𝑝−1_2(𝑝−1)𝜉^(𝑘)_𝑙2𝛾⟶0,as𝑘⟶∞. (47) Thus, for 𝑘 large enough, we have 𝐽𝑡_𝑘𝑢^(𝑘)≥𝐽𝜉^(𝑘)≥𝛿2𝜉^(𝑘)2_𝑙2−_{𝑛∈𝑍𝑚}𝛾_𝑛𝐹_𝑛𝜉_𝑛^(𝑘)≥2𝛿𝑀−_{𝑛∈𝑍𝑚}𝛾_𝑛𝐹_𝑛𝜉_𝑛^(𝑘). (48)
By (47), (48), and 𝑀>𝑐/2𝛿, we have lim_𝑘→∞𝐽𝑡_𝑘𝑢^(𝑘)≥2𝛿𝑀>𝜃𝑐. (49) Noting that 𝐽(0)=0 and 𝐽(𝑢^(𝑘))→𝑐, as 𝑘→∞, then 0<𝑡_𝑘<1 when 𝑘 is big enough. Thus, ⟨𝐽^′(𝑡_𝑘𝑢^(𝑘)),𝑡_𝑘𝑢^(𝑘)⟩=0. In view of (𝐻3), it follows that 𝐽𝑡_𝑘𝑢^(𝑘)=𝐽𝑡_𝑘𝑢^(𝑘)−12𝐽^′𝑡_𝑘𝑢^(𝑘),𝑡_𝑘𝑢^(𝑘)=_{𝑛∈𝑍𝑚}𝛾_𝑛12𝑓_𝑛𝑡_𝑘𝑢_𝑛^(𝑘)𝑡_𝑘𝑢_𝑛^(𝑘)−𝐹_𝑛𝑡_𝑘𝑢_𝑛^(𝑘)≤𝜃_{𝑛∈𝑍𝑚}𝛾_𝑛12𝑓_𝑛𝑢_𝑛^(𝑘)𝑢_𝑛^(𝑘)−𝐹_𝑛𝑢_𝑛^(𝑘)=𝜃𝐽𝑢^(𝑘)−12𝐽^′𝑢^(𝑘),𝑢^(𝑘). (50) By (50), we have lim_𝑘→∞𝐽𝑡_𝑘𝑢^(𝑘)≤𝜃𝑐. (51) This contradicts (49), so {𝑢^(𝑘)} is bounded in 𝑙².
Second, we show that there exists a convergent subsequence of {𝑢^(𝑘)}. In fact, there exists a subsequence, still denoted by the same notation, such that 𝑢^(𝑘)⇀𝑢,in𝑙². (52) By Lemma 2, we have 𝑢^(𝑘)⟶𝑢,in𝑙²_𝛾. (53) By direct calculation, we obtain 𝑢^(𝑘)−𝑢²_𝑙2𝜔=𝐽^′𝑢^(𝑘)−𝐽^′(𝑢),𝑢^(𝑘)−𝑢+_{𝑛∈𝑍𝑚}𝛾_𝑛𝑓_𝑛𝑢_𝑛^(𝑘)−𝑓_𝑛𝑢_𝑛𝑢_𝑛^(𝑘)−𝑢_𝑛≤𝐽^′𝑢^(𝑘)−𝐽^′(𝑢),𝑢^(𝑘)+−𝑢_{𝑛∈𝑍𝑚}𝛾_𝑛𝜖𝑢_𝑛^(𝑘)+𝑢_𝑛+𝑐₁𝑢_𝑛^{(𝑘)𝑝−1}+𝑢_𝑛^𝑝−1×𝑢_𝑛^(𝑘)−𝑢_𝑛≤𝐽^′𝑢^(𝑘)−𝐽^′(𝑢),𝑢^(𝑘)+−𝑢𝛾^1/2𝜖𝑢^(𝑘)_𝑙2+‖𝑢‖_𝑙²+𝑐₁𝑢^(𝑘)_𝑙^𝑝−1_2(𝑝−1)+‖𝑢‖_𝑙^𝑝−1_2(𝑝−1)×𝑢^(𝑘)−𝑢_𝑙2𝛾. (54) Therefore, combining (11), (24), (52), (53), (54), and the boundedness of {𝑢^(𝑘)}, it is clear that lim_𝑘→∞𝑢^(𝑘)−𝑢_𝑙2=0 (55) and this means 𝐽 satisfies (𝐶)_𝑐 condition. The proof is complete.

Now, we are ready to prove Theorem 5.

Proof of Theorem 5. Let 𝑎, 𝜌, and 𝑒∈𝑙² be obtained in Lemma 10.
Since 𝐽(𝑡𝑒)→−∞ as |𝑡|→∞, there exists a real number 𝑡₀ such that 𝑡₀𝑒_𝑙²>𝜌,𝐽𝑡₀𝑒<0. (56) Immediately, we obtain max𝐽(0),𝐽𝑡₀𝑒=0<𝑎≤inf_{‖𝑢‖=𝜌}𝐽(𝑢). (57) Now that we have verified all assumptions of Lemma 4, we know 𝐽 possesses a (𝐶)_𝑐 sequence {𝑢_𝑗}⊂𝑙² for the mountain pass level 𝑐≥𝑎 with 𝑐=inf_ℎ∈Γmax_[]𝑠∈0,1𝐽(ℎ(𝑠)), (58) where Γ=ℎ∈𝐶[0,1],𝑙²∶ℎ(0)=0,ℎ(1)=𝑡₀𝑒. (59)
By Lemma 11, {𝑢_𝑗} has a convergent subsequence {𝑢_𝑗𝑚} such that 𝑢_𝑗𝑚→𝑢 as 𝑗_𝑚→+∞ for some bounded 𝑢∈𝑙². Since 𝐽∈𝐶¹(𝑙²,𝑅), we have 𝐽𝑢_𝑗𝑚(𝑢),𝑢⟶𝐽1+_{𝑗𝑚𝑙2}𝐽^′𝑢_𝑗𝑚⟶1+‖𝑢‖_𝑙²𝐽^′(𝑢), (60) as 𝑗_𝑚→+∞. By the uniqueness of limit and the fact that 𝑢 is bounded, we obtain that 𝑢 is a nontrivial critical point of 𝐽 as the corresponding critical value 𝑐≥𝑎>0. Hence, (1) has at least one nontrivial solution 𝑢 in 𝑙².
Finally, we show that 𝑢={𝑢_𝑛} satisfies (28). In fact, similar to [39], for 𝑛∈𝑍, let 𝑤_𝑛=−𝛾_𝑛𝑓_𝑛𝑢_𝑛𝑢_𝑛,𝑢_𝑛≠0,0,𝑢_𝑛=0; (61) then 𝐿𝑢_𝑛=𝜔𝑢_𝑛, (62) where 𝐿𝑢_𝑛=𝐿𝑢_𝑛+𝑤_𝑛𝑢_𝑛. (63) Clearly, lim_|𝑛|→∞𝑤_𝑛=0. Thus, the multiplication by 𝑤_𝑛 is a compact operator in 𝑙², which implies that 𝜎_ess𝐿=𝜎_ess(𝐿), (64) where 𝜎_ess stands for the essential spectrum. Equation (62) means that 𝑢={𝑢_𝑛} is an eigenfunction that corresponds to the eigenvalue of finite multiplicity 𝜔∉𝜎_ess(𝐿) of the operator 𝐿. Equation (28) follows from the standard theorem on exponential decay for such eigenfunctions [1]. Now the proof of Theorem 5 is complete.

Conflict of Interests

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

Acknowledgments

This work is supported by the Program for Changjiang Scholars and Innovative Research Team in University (no. IRT1226), the National Natural Science Foundation of China (no. 11171078), the Specialized Fund for the Doctoral Program of Higher Education of China (no. 20114410110002), and the Project for High Level Talents of Guangdong Higher Education Institutes.