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

Authors: Defang Ma; Zhan Zhou
Journal: Abstract and Applied Analysis
 (2012)

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## 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.


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## Body

## 1. Introduction

Assume thatm is a positive integer. Consider the following difference equation in infinite m dimensional lattices,
(1.1)Lun+vnun-ωun=σf(n,un),n∈ℤm,
where σ=±1, n=(n1,n2,…,nm)∈ℤm,  {un} is a real valued sequence,  ω∈ℝ, L is a Jacobi operator [1] given by
(1.2)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),
where {ain}(i=1,2,…,m) and {bn} are real valued and bounded sequences.We assume thatf(n,0)=0 for n∈ℤm, then un=0 is a solution of (1.1), which is called the trivial solution. As usual, we say that a solution u={un} of (1.1) is homoclinic (to 0) if
(1.3)lim|n|→∞un=0,
where |n|=|n1|+|n2|+⋯+|nm| is the length of multiindex n. In addition, if un≢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:
(1.4)iψ˙n+Δψn-vnψn+σf(n,ψn)=0,n∈ℤm,
where
(1.5)Δψn=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)
is the discrete Laplacian in m spatial dimension. Moreover, assume that the nonlinearity f(n,u) is gauge invariant, that is,
(1.6)f(n,eiθu)=eiθf(n,u),θ∈ℝ.
Since solitons are spatially localized time-periodic solutions and decay to zero at infinity. Thus ψn has the form
(1.7)ψn=une-iωt,lim|n|→∞ψn=0,
where {un} is a real-valued sequence, and ω∈ℝ is the temporal frequency. Then (1.4) becomes
(1.8)-Δun+vnun-ωun=σf(n,un),n∈ℤm,
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 ain≡-1(i=1,2,…,m), bn≡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].Whenm=1, vn≡0, and {an}, {bn}, 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 {an},{bn}, 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 {vn} 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 Section2, 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(2.1)lp≡lp(ℤm)={u={un}n∈ℤm:∀n∈ℤm,un∈ℝ,∥u∥lp=(∑n∈ℤm‍|un|p)1/p<∞}.
Then the following embedding between lp spaces holds:
(2.2)lq⊂lp,∥u∥lp≤∥u∥lq,1≤q≤p≤∞.Assume the following condition on{vn} holds.(
V
1
)  the discrete potential V={vn}n∈ℤm satisfies
(2.3)lim|n|→∞vn=∞.Let(2.4)H=L+V.
Since the operator L is bounded and self-adjoint in the space l2(ℤm) 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 vn>∥L∥ for all n∈ℤm. Then the operator H is an unbounded positive self-adjoint operator in l2(ℤm).Define the space(2.5)E:={u∈l2(ℤm):H1/2u∈l2(ℤm)}.
Then E is a Hilbert space equipped with the norm
(2.6)∥u∥=∥H1/2u∥l2(Zm).Since(V1) holds, we see that the spectrum σ(H) is discrete and let λ1 be the smallest eigenvalue of H, that is
(2.7)λ1=infσ(H).Now, we present the following basic hypotheses in order to establish the main results in this paper:, and there existsa>0,p∈(2,∞) such that
(2.8)|f(n,u)|≤a(1+|u|p-1),∀n∈ℤm,u∈ℝ.(
f
2
)
lim
|
u
|
→
0
f
(
n
,
u
)
/
u
=
0 uniformly for n∈ℤm.(
f
3
)
lim
|
u
|
→
∞
F
(
n
,
u
)
/
u
2
=
+
∞ uniformly for n∈ℤm, where F(n,u) is the primitive function of f(n,u), that is,
(2.9)F(n,u)=∫0u‍f(n,t)dt.(
f
4
)
f
(
n
,
u
)
/
u is  increasing  in u>0 and  decreasing  in  u<0, for all n∈ℤm.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)
Ifσ=-1,  ω≤λ1, (1.1) has no nontrivial solution in E.(2)
Ifσ=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 constantsC and α such that
(2.10)|un|≤Ce-α|n|,n∈ℤm.Theorem 2.2.
Assumef(n,u) is odd in u for each n∈ℤm, and that conditions (V1),(f1)–(f4) hold. Then we have the following conclusions.(1)
Ifσ=-1,  ω≤λ1, (1.1) has no nontrivial solution in E.(2)
Ifσ=1,  ω<λ1,  (1.1) has infinitely many solutions {u(k)}k=1∞ in E satisfying
(2.11)12(Hu(k),u(k))-12ω(u(k),u(k))-∑n∈Zm‍F(n,un(k))→∞,ask→∞.(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
(2.12)Hun-ωun-σγnf(un)=0,
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∈ℤm,
(2.13)γ_≤γn≤γ-.
The nonlinearityf is odd and satisfies
(A2) there are two positive constants C1, C2, and 2<p<∞ such that
(2.14)|f(u)|≤C1(1+|u|p-1),(2.15)|f(u)-f(v)|≤C2(1+|u|p-2+|v|p-2)|u-v|.
(A3) limu→0(f(u)/|u|)=0.
(A4) there is a 2<q<∞ such that
(2.16)0<(q-1)f(u)u≤f′(u)u2,∀u≠0.
Then we have the following conclusions.(1)
Ifσ=-1,  ω≤λ1, (2.12) has no nontrivial solution.(2)
Ifσ=1,  ω<λ1, (2.12) has at least a pair of nontrivial solutions ±u in l2(ℤm).(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
(2.17)L=-Δ,f(n,u)=γnf(u).
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
(2.18)f(u)={0,u=0,u1-ln|u|,0<|u|≤1,u(1+ln|u|),|u|>1,
then f does not satisfy (A4). However, if we let f(n,u)=γnf(u) in (1.1), where {γn} 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 operatorL is bounded in l2(ℤm), the following two norms are equivalent in the Hilbert space E(2.19)∥u∥~∥V1/2u∥l2(ℤm).
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 lp(ℤm) is compact, denote the best embedding constant cp=max∥u∥lp=11/∥u∥.Consider the functionJ defined on E by
(2.20)J(u)=12((H-ω)u,u)-σ∑n∈Zm‍F(n,un).
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 ofJ we have the following formula:
(2.21)(J′(u),v)=((H-ω)u,v)-σ∑n∈Zm‍f(n,un)vn,∀v∈E.Definition 2.6 (see [22, 23]).
LetE 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 {un}⊂E such that J(un)→c and ∥J'(un)∥(1+∥un∥)→0asn→∞, has a convergent subsequence.
LetBr be the open ball in H with radius r and center 0, and let ∂Br 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.
LetE be a real Banach Space, J∈C1(E,ℝ) satisfies the (C)c condition for any c>0,J(0)=0, and
(
J
1
) There exist ρ,σ>0 such that J∣∂Bρ≥α.
(
J
2
) There exist e∈E∖Bρ such that J(e)≤0.
ThenJ has a critical value c≥α.In order to prove Theorem2.2, we shall use the following fountain theorem [23, 25, 26]. Let E be a real Banach Space with the norm ∥·∥ and E=⨁j∈ℕXj¯ with dim Xj<∞ for any j∈ℕ. Set Yk=⨁j=0kXj and Zk=⨁j=k∞Xj¯.Lemma 2.8.
LetJ∈C1(E,ℝ) be even. If, for each sufficiently large k∈ℕ, there exists ρk>γk>0 such that
(
B
1
)  ak:=maxu∈Yk,∥u∥=ρkJ(u)≤0.
(
B
2
)
b
k
:
=
inf
u
∈
Z
k
,
∥
u
∥
=
γ
k
J
(
u
)
→
∞
,
k
→
∞.
(
B
3
) 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 thatσ=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,
(3.1)|f(n,u)|≤ϵ|u|+c1|u|p-1
for all n∈ℤm and u∈ℝ. This, together with the mean value theorem, leads to
(3.2)|F(n,u)|=|F(n,u)-F(n,0)|=|∫01‍f(n,su)uds|≤ϵ2|u|2+c1p|u|p.
By (3.2) and the Hölder inquality, it follows that
(3.3)J(u)=12((H-ω)u,u)-∑n∈ℤm‍F(n,un)≥12(λ1-ω)∥u∥22-(ϵ2∑n∈ℤm‍|un|2+c1p∑n∈ℤm‍|un|p)≥14(λ1-ω)∥u∥22-c1p∥u∥2p.
Noting that p>2, we obtain the following estimate:
(3.4)J(u)≥18(λ1-ω)ρ2≡α>0,∀∥u∥=ρ,
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∈ℝm,|u|≥δ, we have
(3.5)F(n,u)≥M|u|2.
Notice that, from (f2) and (f4), it is easy to get that
(3.6)F(n,u)>0,∀u≠0.
Lete∈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
(3.7)∑|n|≤N‍en2≥12∥e∥22.
Let
(3.8)A*={n∈ℤm:|n|≤N,en≠0}.
Taking t large enough, such that |ten|>δ for all n∈A*, then, in view of (3.5)–(3.7), we have
(3.9)J(te)=12(λ1-ω)t2∥e∥22-∑n∈ℤm‍F(n,ten)≤12(λ1-ω)t2∥e∥22-∑n∈A*‍F(n,ten)≤12(λ1-ω)t2∥e∥22-Mt2∑n∈A*‍en2≤12(λ1-ω-M)t2∥e∥22.
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 thatσ=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,
(3.10)J(u(k))→c,∥J′(u(k))∥(1+∥u(k)∥)→0,ask→∞.
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
(3.11)α(k)⇀αinE,(3.12)α(k)→α,inlq(ℤm),forq≥2.Case1 (α≠0). By J(u(k))=c+o(1), where o(1)→0 as k→∞, we have
(3.13)c+o(1)=J(u(k))=12∥u(k)∥2-12ω∥u(k)∥22-∑n∈ℤm‍F(n,un(k)).
Noticing that ∥u∥2=(Hu,u)≥λ1∥u∥22, we divide both sides of (3.13) by ∥u(k)∥2 and get
(3.14)∑n∈ℤm‍F(n,un(k))∥u(k)∥2≤(12+|ω|2λ1)-c+o(1)∥u(k)∥2<+∞.
Let Ω={n∈ℤm:α(n)≠0}, then it follows from (3.12) that
(3.15)un(k)=αn(k)∥u(k)∥→+∞,ask→∞,forn∈Ω.
In view of (f3), we have
(3.16)limk→∞F(n,un(k))∥u(k)∥2=limk→∞F(n,un(k))|un(k)|2|αn(k)|2→+∞,forn∈Ω.
Therefore,(3.17)∑n∈ℤm‍F(n,un(k))∥u(k)∥2=(∑n∈Ω‍+∑n∉Ω‍)F(n,un(k))∥u(k)∥2≥∑n∈Ω‍F(n,un(k))∥u(k)∥2→+∞.
This contradicts (3.14).
Case  2 (α=0). We define
(3.18)J(tku(k)):=maxt∈[0,1]J(tu(k)).
For any M>4, let k be large enough such that ∥u(k)∥>M  and  α¯(k):=2M1/2u(k)/∥u(k)∥=2M1/2α(k).
By (3.2), (3.12), and α=0, it is clear that
(3.19)∑n∈ℤm‍F(n,α¯nk)≤ϵ2∥α¯∥22+c1p∥α¯∥2p→0,ask→∞.
Thus, for k large enough
(3.20)J(tku(k))≥J(α¯n(k))=12∥α¯(k)∥2-12ω∥α¯(k)∥22-∑n∈ℤm‍F(n,α¯n(k))≥12∥α¯(k)∥2-12ω0∥α¯(k)∥22-∑n∈ℤm‍F(n,α¯n(k))≥12(1-ω0λ1)∥α¯(k)∥2-∑n∈ℤm‍F(n,α¯n(k))≥(1-ω0λ1)M,
where ω0=max{ω,0}<λ1.
This implies thatlimk→∞J(tku(k))=∞. Since J(0)=0 and J(u(k))→cas  k→∞, J(tu(k)) attains its maximum at tk∈(0,1) for large k. Thus, (J'(tku(k)),u(k))=0.
On the other hand, from(f4), we have that
(3.21)G(n,s)≤G(n,t),∀0≤s<tort<s≤0,n∈ℤm,
where G(n,t)=f(n,t)t/2-F(n,t). In fact, for 0<s<t  ort<s<0, we have
(3.22)G(n,t)-G(n,s)=tf(n,t)2-sf(n,s)2-∫st‍f(n,τ)dτ≥tf(n,t)2-sf(n,s)2-f(n,t)t∫st‍τdτ≥s22(f(n,t)t-f(n,s)s)>0.
If s=0, (3.21) is obvious.
By (3.10) and (3.21), we have
(3.23)J(tku(k))-12(J′(tku(k)),tku(k))=∑n∈ℤm‍(12f(n,tkun(k))tkun(k)-F(n,tkun(k)))≤∑n∈ℤm‍(12f(n,un(k))un(k)-F(n,un(k)))=J(u(k))-12(J′(u(k)),u(k))→c,ask→∞.
But (3.20) implies that
(3.24)J(tku(k))-12(J′(tku(k)),tku(k))=J(tku(k))→∞.
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≤∞,
(3.25)u(k)→u,inlq(ℤm).
By a straightforward calculation, we have
(3.26)∥u(k)-u∥2-ω∥u(k)-u∥22=(J′(u(k))-J′(u),(u(k)-u))+∑n∈ℤm‍(f(n,un(k))-f(n,un))(un(k)-un).
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
(3.27)∑n∈ℤm‍(f(n,un(k))-f(n,un))(un(k)-un)≤∑n∈ℤm‍[ϵ(|un(k)|+|un|)+c1(|un(k)|p-1+|un|p-1)](un(k)-un)≤ϵ(∥u(k)∥2+∥u∥2)∥u(k)-u∥2+c1(∥u(k)∥pp-1+∥u∥pp-1)∥u(k)-u∥p.
Therefore, combining (3.25) and the boundedness of {u(k)}, the above inequality implies
(3.28)∑n∈ℤm‍(f(n,un(k))-f(n,un))(un(k)-un)→0,ask→∞.
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 Theorem2.1.
(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
(3.29)(J′(u),u)=((H-ω)u,u)-σ∑n∈ℤm‍f(n,un)un≥∑n∈ℤm‍f(n,un)un>0.
This is a contradiction, so the conclusion holds.
(2) By Lemma3.1, 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) Assumew is a nontrivial solution obtained in (2). Similar to [17], let Θ={θn≡-σf(n,wn)/wn} and the operator G defined by (Gu)n=(Hu)n+θnun for u∈E, then (1.1) is equivalent to
(3.30)Gu=ωu.
Since w∈E, we have lim|n|→∞θn=0. The multiplication operator Θ is compact in E⊂l2(ℤm), and this implies that
(3.31)σess(G)=σess(H)=ϕ.
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 Theorem2.1.Assume thati∈ℤm. Define e(i)={en(i)} by
(3.32)en(i)={1,n=i,0,n≠i.LetXj=span{e(n):|n|=j} for j∈ℕ={0,1,2,…}, Yk=⨁j=0kXj, and Zk=⨁j=k∞Xj¯, then we haveLemma 3.3.
Suppose thatσ=1,ω<λ1, (V1),(f1)–(f3) hold. Then there exists ρk>γk>0 such that(i)
bk:=infu∈Zk,∥u∥=γkJ(u)→∞,ask→∞,(ii)
ak:=maxu∈Yk,∥u∥=ρkJ(u)≤0.Proof.
(i) It follows from (3.2) that, for any u∈Zk(3.33)J(u)=12(Hu,u)-12ω(u,u)-∑n∈ℤm‍F(n,un)≥12∥u∥2-12(ω+ϵ)∥u∥22-c1p∥u∥pp≥12[1-(|ω|+ϵ)βk2]∥u∥2-c1pβkp∥u∥p,
where βk=supu∈Zk,∥u∥=1∥u∥2. Since for u∈Zk,
(3.34)∥u∥2=(Hu,u)=(Lu,u)+(Vu,u)≥-∥L∥∥u∥22+∑|n|≥k‍vnun2≥inf|n|≥k(vn-∥L∥)∥u∥22,
we see that βk→0 as k→∞. Thus,
(3.35)bk:=infu∈Zk,∥u∥=γkJ(u)≥infu∈Zk,∥u∥=γk12[1-(|ω|+ϵ)βk2]∥u∥2-c1pβkp∥u∥p=14[1-(|ω|+ϵ)βk2]γk2,
where γk=[p(1-|ω|βk2-ϵβk2)/4c1βkp]1/(p-2). Notice that p>2 and βk→0 as k→∞, so we have bk→∞,  as k→∞.
(ii) For anyk∈ℕ, let the dimension of Yk be χk and vk*=max|j|≤k{vj}. Assume that  M=∥L∥+vk*-ω, then from (3.5), there exists a  δk>0  such that F(n,u)≥M|u|2  for |u|≥δk. Thus, for u∈Yk,
(3.36)J(u)=12((H-ω)u,u)-∑|n|≤k‍F(n,un)=12((L+V-ω)u,u)-∑|un|≤δkF(n,un)-∑|un|>δkF(n,un)≤12(∥L∥+vk*-ω)∥u∥22-M∑|un|>δkun2≤12(∥L∥+vk*-ω)∥u∥22-M∑|n|≤k‍un2+M∑|un|≤δkun2≤12(∥L∥+vk*-ω-2M)∥u∥22+Mδk2χk=-12(∥L∥+vk*-ω)∥u∥22+(∥L∥+vk*-ω)δk2χk.
Taking ρk sufficiently large, we have,
(3.37)ak:=maxu∈Yk,∥u∥=ρkJ(u)≤0.
The proof is completed.Proof of Theorem2.2.
The proofs for (1) and (3) are similar to that of (1) and (3) in Theorem2.1, 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(ℤm). This completes the proof.



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*Source: 703596-2012-10-17.xml*