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

Authors: Genghong Lin; Zhan Zhou
Journal: Abstract and Applied Analysis
 (2014)

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## 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.


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## Body

## 1. Introduction

Assume thatm is a positive integer. Consider the following difference equation in infinite higher dimensional lattices:
(1)Lun-ωun=σγnfn(un),n∈Zm,
where fn(u) is continuous in u, σ=±1, n=(n1,n2,…,nm)∈Zm, γ={γn} is a positive real valued sequence, ω∈R, and L is a Jacobi operator [1] given by
(2)Lun=a1(n1,n2,…,nm)u(n1+1,n2,…,nm)+a1(n1-1,n2,…,nm)u(n1-1,n2,…,nm)+a2(n1,n2,…,nm)u(n1,n2+1,…,nm)+a2(n1,n2-1,…,nm)u(n1,n2-1,…,nm)+⋯+am(n1,n2,…,nm)u(n1,n2,…,nm+1)+am(n1,n2,…,nm-1)u(n1,n2,…,nm-1)+b(n1,n2,…,nm)u(n1,n2,…,nm);
here, {ain}(i=1,2,…,m) and {bn} are real valued bounded sequences.Assume thatfn(0)=0 for n∈Zm; then {un}={0} is a solution of (1), which is called the trivial solution. As usual, we say that u={un}, a solution of (1), is homoclinic (to 0) if
(3)lim⁡|n|→∞un=0,
where |n|=|n1|+|n2|+⋯+|nm| is the length of multi-index n. In addition, if {un}≠{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
(4)iψ˙n=-Δψn-σγnfn(ψn),n∈Zm,
where σ=±1 and
(5)Δψn=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)
is the discrete Laplacian in m spatial dimension. Typical representatives of power nonlinearities are
(6)fn(u)=ln|u|pu,ln,p>0.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,
(7)fn(eiθu)=eiθfn(u),θ∈R,
and, in addition, fn(u)≥0 for u≥0 for n∈Zm.Making use of the standing wave ansatz,(8)ψn=une-iωt,lim⁡|n|→∞ψn=0,
where {un} is a real valued sequence and ω∈R is the temporal frequency. Then (4) becomes
(9)-Δun-ωun=σγnfn(un),n∈Zm,
and (3) holds. This is an equation of the form (1) with ain=-1(i=1,2,…,m) and bn=2m.Whenfn(u) has the form of (6), the homoclinic solutions of (9) were obtained by Karachalios in [2] by assuming that γ∈lρ, ρ=(q-1)/(q-2), for some q>2. We note that γ∈lρ implies that lim⁡|n|→∞γn=0. Moreover, (6) satisfies the classical Ambrosetti-Rabinowitz superlinear condition [3], and fn(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 offn(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|→∞γn=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 operatorL 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{γn} converges to zero at infinity; that is, lim⁡|n|→∞γn=0. After that, we prove a compact inclusion between ordinary sequence spaces l2 and weighted sequence spaces lγ2 (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
(10)lp=lp(Zm)={=(∑n∈Zm‍|un|p)1/pu={un}n∈Zm:∀n∈Zm,un∈R,∥u∥lp=(∑n∈Zm‍|un|p)1/p<∞}.
Then the following embedding between lp spaces holds:
(11)lq⊂lp,∥u∥lp≤∥u∥lq,1≤q≤p≤∞.
For p=2, we get the usual Hilbert space of square-summable sequences, with the real scalar product
(12)(u,v)l2=∑n∈Zm‍unvn,u,v∈l2.For a positive real valued bounded sequenceγ={γn:0<γn≤γ¯<∞}n∈Zm, we define the weighted sequence spaces lγ2:
(13)lγ2={(∑n∈Zm‍γn|un|2)1/2u={un}n∈Zm:∀n∈Zm,un∈R,∥u∥lγ2=(∑n∈Zm‍γn|un|2)1/2<∞}.
It is not hard to see that lγ2 is a Hilbert space, with the scalar product
(14)(u,v)lγ2=∑n∈Zm‍γnunvn,u,v∈lγ2.For a certain class of weightγ, we have the following lemmas, which will play a crucial role in our analysis.Lemma 1.
Letκ={κn:|κn|≤κ¯<∞}n∈Zm be a multiplication operator from lγ2 to lγ2 defined by κu={κnun}n∈Zm. If lim⁡|n|→∞κn=0, then the operator κ is compact.Proof of Lemma1.
Let(15)Λ={κu:∥u∥lγ2≤1}.
We only need to prove that Λ is precompact in lγ2. By assumption, for any ɛ>0, there exists N>0 such that |κn|≤ɛ for any |n|>N. Define a cutting sequence χ={χn} by
(16)χn={1,|n|≤N,0,|n|>N.
Denote by χc=1-χ the anticutting sequence. Then for any κu∈Λ(17)∥χcκu∥lγ22=∑|n|>N‍γn|κnun|2≤ɛ2,∥χκu∥lγ22=∑|n|≤N‍γn|κnun|2≤κ¯2.
For arbitrary ɛ>0 and Λɛ={χκu:∥u∥lγ2≤1} finite-dimensional and bounded, we know that Λ is precompact. The proof is complete.Lemma 2.
One assumes positive sequence of real numbersγ with lim⁡|n|→∞γn=0. Then l2↪lγ2 with compact inclusion.Proof of Lemma2.
Note that∥u∥lγ2≤γ¯∥u∥l2 for any u∈l2 and γ1/2 is compact. Thus, l2↪lγ2 with compact inclusion by Lemma 1. The proof is complete.Remark 3.
Karachalios [2] proved l2↪lγ2 with compact inclusion assuming that γ∈lρ, ρ=(q-1)/(q-2), for some q>2. Note that γ∈lρ implies that lim⁡|n|→∞γn=0. Thus, we find that Lemma 2 improves Lemma 2.1 in [2].On the Hilbert spacel2, we consider the functional
(18)J(u)=12((L-ω)u,u)l2-σ∑n∈Zm‍γnFn(un),
where
(19)Fn(u)=∫0u‍fn(s)ds
is the primitive function of fn(u). Then J∈C1(l2,R) and
(20)〈J′(u),v〉=((L-ω)u,v)l211111111111-σ∑n∈Zm‍γnfn(un)vn,u,v∈l2.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.Letδ be the distance from ω to the spectrum σ(L); that is,
(21)δ=β-ω.
Then, we have
(22)((L-ω)u,u)≥δ∥u∥l22,u∈l2.
We also consider a norm in l2 defined by
(23)∥u∥lω2=[((L-ω)u,u)]1/2,u∈l2.
Since
(24)δ∥u∥l22≤∥u∥lω22≤∥L-ω∥∥u∥l22,u∈l2,
norm (23) is an equivalent norm with the usual one of l2.In order to obtain the existence of critical points ofJ on l2, we cite some basic notations and some known results from critical point theory.LetH 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.LetJ∈C1(H,R). A sequence {uj}⊂H is called a (C)c sequence for J if J(uj)→c for some c∈R and (1+∥uj∥)∥J′(uj)∥ as j→∞. We say J satisfies the (C)c condition if any (C)c sequence for J possesses a convergent subsequence.LetBr be the open ball in H with radius r and center 0, and let ∂Br denote its boundary. The following lemma is taken from [37].Lemma 4 (Mountain Pass Lemma).
IfJ∈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∥=rJ(u), then there exists a (C)c sequence {un} for the mountain pass level c which is defined by
(25)c=inf⁡h∈Γmax⁡s∈[0,1]J(h(s)),
where
(26)Γ={h∈C([0,1],H):h(0)=0,h(1)=e}.


## 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 thatσ=1, ω∈(-∞,β), and the following conditions hold. (H1)
fn(u) is continuous in u, fn(u)=o(u) as u→0 uniformly for n∈Zm.(H2)
There existb>0, p>2 such that
(27)|fn(u)|≤b(1+|u|p-1)
uniformly for n∈Zm and u∈R.(H3)
There exists someθ≥1 such that θGn(u)≥Gn(tu), for n∈Zm, u∈R, and t∈[0,1], where Gn(u)=(1/2)fn(u)u-Fn(u).(H4)
Fn(u)≥0 for u∈R, and lim⁡|u|→∞(Fn(u)/u2)=∞, uniformly for n∈Zm.(H5)
Positive real valued sequenceγ={γn}n∈Zm with lim⁡|n|→∞γn=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
(28)|un|≤Ce-τ|n|,n∈Zm.Theorem5 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 thatσ=-1, ω≤β, and γnfn(u)u>0 when u≠0 for all n∈Zm. Then (1) has no nontrivial solutions in l2.Proof of Proposition6.
By way of contradiction, we assume that (1) has a nontrivial solution u={un}∈l2. Then u is a nonzero critical point of J, and
(29)〈J′(u),u〉=((L-ω)u,u)l2-σ∑n∈Zm‍γnfn(un)un≥∑n∈Zm‍γnfn(un)un>0.
This is a contradiction as 〈J′(u),u〉=0, so the conclusion holds.Remark 7.
It is easy to see that the functionfn defined by
(30)fn(u)=|u|2λu,
where λ>0 and γ∈lρ, 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(H¯3): fn(u)/|u| is nondecreasing with respect to |u|. We want to point out that condition (H3) is equivalent to (H¯3) when θ=1 and (H3) gives “better monotony” when θ>1, since (H¯3) implies (H3) (see [38]). Moreover, we can find that fn(u)=5uln⁡(1+u2)+9sinu satisfies (H3) but not (H¯3) for some θ≥100.Remark 9.
As we know, the condition(H¯4)
f(u)u>qF(u)>0forsomeq>2andu≠0is often called Ambrosetti-Rabinowitz superlinear condition [3]. Clearly, (H¯4) implies (H4). Actually, (H4) is more general than (H¯4). Let {an} be a positive sequence, and fn(u)=anuln⁡(1+|u|). Then fn satisfies (H4). However, fn does not satisfy (H¯4).The proof of Theorem5 is based on a direct application of the following lemmas. The key points read as follows.Lemma 10.
Assume that the conditions of Theorem5 hold; then one has the following. (J1)
There exist two constantsa>0 and ρ>0 such that J|∂Bρ≥a.(J2)
There exists ane∈l2 such that J(te)→-∞ as |t|→∞.Proof of Lemma10.
Letγ¯=max⁡n∈Zm{γn} and ϵ=δ/2γ¯. By (H1) and (H2), there exists c1>0, such that
(31)|fn(u)|≤ϵ|u|+c1|u|p-1
for all n∈Zm and u∈R, and (31) implies that
(32)|Fn(u)|≤ϵ2|u|2+c1p|u|p.
By (32) and the Hölder inequality, we have
(33)J(u)=12((L-ω)u,u)l2-∑n∈Zm‍γnFn(un)≥δ2∥u∥l22-γ¯∑n∈Zm‍(ϵ2|u|2c1p|u|p)≥δ4∥u∥l22-c1γ¯p∥u∥l2p.
Since p>2, we have
(34)J(u)≥δρ28=a>0for∥u∥l2=ρ,
where ρ=(δp/8c1γ¯)1/(p-2).
Lete={en}∈l2 be the eigenvector of L corresponding to the eigenvalue β; that is to say, Le=βe. There exists N>0, such that
(35)∑|n|≤N‍en2≥12∥e∥l22.
Let
(36)A*={n∈Zm:|n|≤N,en≠0}.
By (H4), for any M>0, there exists η=η(M)>0 such that
(37)Fn(u)≥M|u|2forn∈Zm,|u|≥η.
Taking t large enough, such that ten>η for all n∈A*, then, combining (35), (36), and (37), we have
(38)J(te)=12((L-ω)te,te)l2-∑n∈Zm‍γnFn(ten)≤δ2t2∥e∥l22-∑n∈A*‍γnFn(ten)≤δ2t2∥e∥l22-γ_Mt2∑n∈A*‍en2≤12(δ-γ_M)t2∥e∥l22,
where γ_=min⁡n∈A*{γn}>0. Letting M be large enough, such that δ≤γ_M, we obtain that J(te)→-∞ as |t|→∞. The proof is complete.Lemma 11.
Assume that the conditions of Theorem5 hold; then the functional J satisfies the (C)c condition for any given c∈R.Proof of Lemma11.
Let{u(k)}⊂l2 be a (C)c sequence of J; that is,
(39)kkkkkkkkkkkkkklJ(u(k))⟶c,(1+∥u(k)∥l2)∥J′(u(k))∥l2⟶0,kkkkkkkkkkkkkkkkkkask⟶∞.
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
(40)ξ(k)⇀ξ,inl2,(41)ξ(k)⟶ξ,inlγ2.
Case
1   (ξ≠0). By J(u(k))=c+o(1), where o(1)→0 as k→0, we have
(42)∑n∈Zm‍γnFn(un(k))∥u(k)∥l22=12((L-ω)u(k),u(k))∥u(k)∥l22-c+o(1)∥u(k)∥l22≤∥L-ω∥2-c+o(1)∥u(k)∥l22<∞.
Let B*={n∈Zm:ξn≠0}. Obviously, B* is nonempty. Then, for some n0∈B*, it follows from (41) that
(43)un0(k)=ξn0(k)∥u(k)∥l2⟶∞,ask⟶∞.
Combining (H4) and γn0>0, we have
(44)γn0Fn0(un0(k))∥u(k)∥l22=γn0Fn0(un0(k))|un0(k)|2|ξn0(k)|2⟶∞,ask⟶∞.
However,
(45)∑n∈Zm‍γnFn(un(k))∥u(k)∥l22=∑n≠n0‍γnFn(un(k))∥u(k)∥l22+γn0Fn0(un0(k))∥u(k)∥l22≥γn0Fn0(un0(k))∥u(k)∥l22⟶∞,
as k→∞. This contradicts (42).
Case
2  (ξ=0). Let
(46)J(tku(k))=max⁡t∈[0,1]J(tu(k)).
For any given M>max⁡{4,θc/2δ}, let k be large enough such that ∥u(k)∥l2>M and ξ¯(k)=2M1/2ξ(k). Combining (32), (41), and ξ=0, it is easy to see that
(47)∑n∈Zm‍γnFn(ξ¯n(k))≤ϵ2∥ξ¯(k)∥lγ22+c1γ¯1/2p∥ξ¯(k)∥l2(p-1)p-1∥ξ¯(k)∥lγ2⟶0,11111111111111111111111111111ask⟶∞.
Thus, for k large enough, we have
(48)J(tku(k))≥J(ξ¯(k))≥δ2∥ξ¯(k)∥l22-∑n∈Zm‍γnFn(ξ¯n(k))≥2δM-∑n∈Zm‍γnFn(ξ¯n(k)).
By (47), (48), and M>c/2δ, we have
(49)lim⁡k→∞J(tku(k))≥2δM>θc.
Noting that J(0)=0 and J(u(k))→c, as k→∞, then 0<tk<1 when k is big enough. Thus, 〈J′(tku(k)),tku(k)〉=0. In view of (H3), it follows that
(50)J(tku(k))=J(tku(k))-12〈J′(tku(k)),tku(k)〉=∑n∈Zm‍γn(12fn(tkun(k))tkun(k)-Fn(tkun(k)))≤θ∑n∈Zm‍γn(12fn(un(k))un(k)-Fn(un(k)))=θ(J(u(k))-12〈J′(u(k)),u(k)〉).
By (50), we have
(51)lim⁡k→∞J(tku(k))≤θc.
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
(52)u(k)⇀u,inl2.
By Lemma 2, we have
(53)u(k)⟶u,inlγ2.
By direct calculation, we obtain
(54)∥u(k)-u∥lω22=〈J′(u(k))-J′(u),u(k)-u〉+∑n∈Zm‍γn(fn(un(k))-fn(un))(un(k)-un)≤〈J′(u(k))-J′(u),u(k)-u〉+∑n∈Zm‍γn(ϵ(|un(k)|+|un|)+c1(|un(k)|p-1+|un|p-1))×(un(k)-un)≤〈J′(u(k))-J′(u),u(k)-u〉111111111111111111+γ¯1/2(ϵ(∥u(k)∥l2+∥u∥l2)111111111111111111111111+c1(∥u(k)∥l2(p-1)p-1+∥u∥l2(p-1)p-1))1111111111111111111111111×∥u(k)-u∥lγ2.
Therefore, combining (11), (24), (52), (53), (54), and the boundedness of {u(k)}, it is clear that
(55)lim⁡k→∞∥u(k)-u∥l2=0
and this means J satisfies (C)c condition. The proof is complete.Now, we are ready to prove Theorem5.Proof of Theorem5.
Leta, ρ, and e∈l2 be obtained in Lemma 10.
SinceJ(te)→-∞ as |t|→∞, there exists a real number t0 such that
(56)∥t0e∥l2>ρ,J(t0e)<0.
Immediately, we obtain
(57)max⁡{J(0),J(t0e)}=0<a≤inf⁡∥u∥=ρJ(u).
Now that we have verified all assumptions of Lemma 4, we know J possesses a (C)c sequence {uj}⊂l2 for the mountain pass level c≥a with
(58)c=inf⁡h∈Γmax⁡s∈[0,1]J(h(s)),
where
(59)Γ={h∈C([0,1],l2):h(0)=0,h(1)=t0e}.
By Lemma11, {uj} has a convergent subsequence {ujm} such that ujm→u as jm→+∞ for some bounded u∈l2. Since J∈C1(l2,R), we have
(60)J(ujm)⟶J(u),(1+∥ujm∥l2)J′(ujm)⟶(1+∥u∥l2)J′(u),
as jm→+∞. 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 thatu={un} satisfies (28). In fact, similar to [39], for n∈Z, let
(61)wn={-γnfn(un)un,un≠0,0,un=0;
then
(62)L¯un=ωun,
where
(63)L¯un=Lun+wnun.
Clearly, lim⁡|n|→∞wn=0. Thus, the multiplication by wn is a compact operator in l2, which implies that
(64)σess(L¯)=σess(L),
where σess stands for the essential spectrum. Equation (62) means that u={un} is an eigenfunction that corresponds to the eigenvalue of finite multiplicity ω∉σess(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.



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