# Existence of Homoclinic Orbits for a Class of Asymptoticallyp-Linear Difference Systems with p-Laplacian

Authors: Qiongfen Zhang; X. H. Tang
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

By applying a variant version of Mountain Pass Theorem in critical point theory, we prove the existence of homoclinic solutions for the following asymptoticallyp-linear difference system with p-Laplacian
Δ(|Δu(n-1)|p-2Δu(n-1))+∇[-K(n,u(n))+W(n,u(n))]=0,
where p∈(1,+∞), n∈ℤ, u∈ℝN, K,W:ℤ×ℝN→ℝ are not periodic in n,
and W is asymptotically p-linear at infinity.


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## Body

## 1. Introduction

Consider the followingp-Laplacian difference system:(1.1)Δ(|Δu(n-1)|p-2Δu(n-1))+∇[-K(n,u(n))+W(n,u(n))]=0,n∈Z,
where Δ is the forward difference operator defined by Δu(n)=u(n+1)-u(n), Δ2u(n)=Δ(Δu(n)), p∈(1,+∞), n∈ℤ, u∈ℝN, K,W: ℤ×ℝN→ℝ are not periodic in n, W is asymptotically p-linear at infinity, and K and W are continuously differentiable in x. As usual, we say that a solution u(n) of (1.1) is homoclinic (to 0) if u(n)→0 as n→±∞. In addition, if u(n)≢0, then u(n) is called a nontrivial homoclinic solution.Whenp=2, (1.1) can be regarded as a discrete analogue of the following second-order Hamiltonian system:(1.2)ü(t)+∇[-K(t,u(t))+W(t,u(t))]=0,t∈R.The existence of homoclinic orbits for Hamiltonian systems is a classical problem and its importance in the study of the behavior of dynamical systems has been recognized by Poincaré [1]. If a system has the transversely intersected homoclinic orbits, then it must be chaotic. If it has the smoothly connected homoclinic orbits, then it cannot stand the perturbation and its perturbed system probably produces chaotic phenomenon. For the existence of homoclinic solutions of problem (1.2), one can refer to the papers [2–5].Difference equations usually describe evolution of certain phenomena over the course of time. For example, if a certain population has discrete generations, the size of the(n+1)th generation x(n+1) is a function of the nth generation x(n). In fact, difference equations provide a natural description of many discrete models in real world. Since discrete models exist in various fields of science and technology such as statistics, computer science, electrical circuit analysis, biology, neural network, and optimal control, it is of practical importance to investigate the solutions of difference equations. For more details about difference equations, we refer the readers to the books [6–8].In some recent papers [9–20], the authors studied the existence of periodic solutions and subharmonic solutions of difference equations by applying critical point theory. These papers show that the critical point theory is an effective method to the study of periodic solutions for difference equations. Along this direction, several authors [21–28] used critical point theory to study the existence of homoclinic orbits for difference equations. Motivated by the above papers, we consider the existence of homoclinic orbits for problem (1.1) by using the variant version of Mountain Pass Theorem. Our result is new, which seems not to have been considered in the literature. Here is our main result.Theorem 1.1.
Suppose thatK and W satisfy the following conditions. (K1)
There are two positive constantsb1 and  b2 such that
(1.3)b1|x|p≤K(n,x)≤b2|x|p,∀(n,x)∈Z×RN.(K2)
There is a positive constantb3 such that
(1.4)b3|x|p≤(∇K(n,x),x)≤|∇K(n,x)||x|≤pK(n,x),∀(n,x)∈Z×RN.(W1)
W(n,0)=0,  ∇W(n,x)=o(|x|p-1) as  |x|→0 uniformly for  n∈ℤ.(W2)
There exists a constantR>0 such that
(1.5)|∇W(n,x)||x|p-1≤R,∀n∈Z,x∈RN.(W3)
There exists a functionV∞∈l∞(ℤ,ℝ+) such that
(1.6)lim⁡|x|→∞|∇W(n,x)-V∞(n)|x|p-2x||x|p-1=0uniformly forn∈Z,inf⁡ZV∞(n)>max⁡{1,pb2}.(W4)
W̃(n,x)=(∇W(n,x),x)-pW(n,x),
(1.7)lim⁡|x|→∞W̃(n,x)=+∞uniformlyforn∈Z,
and for any fixed 0<c1<c2<+∞,
(1.8)inf⁡n∈Z,c1≤|x|≤c2W̃(n,x)|x|p>0.Then problem (1.1) has at least one nontrivial homoclinic solution.Remark 1.2.
The functionW(n,x) in this paper is asymptotically p-linear at infinity. The behavior of the gradient of W(n,x) at infinity is like that of a function V∞(n)|x|p-2x, where V∞(n) is a real function but not a matrix function. To the best of our knowledge, similar results of this kind of p-Laplacian difference systems with asymptotically p-linear W(n,x) at infinity cannot be found in the literature. From this point, our result is new.


## 2. Preliminaries

Let(2.1)S={{u(n)}n∈Z:u(n)∈RN,n∈Z},E={u∈S:∑n∈Z[|Δu(n-1)|p+|u(n)|p]<+∞},
and for u∈E, let (2.2)‖u‖={∑n∈Z[|Δu(n-1)|p+|u(n)|p]<+∞}1/p.
Then E is a uniform convex Banach space with this norm. As usual, for 1≤p<+∞, let (2.3)lp(Z,RN)={u∈S:∑n∈Z|u(n)|p<+∞},l∞(Z,RN)={u∈S:sup⁡n∈Z|u(n)|<+∞},
and their norms are given by (2.4)‖u‖lp=(∑n∈Z|u(n)|p)1/p,∀u∈lp(Z,RN),‖u‖∞=sup⁡{|u(n)|:n∈Z},∀u∈l∞(Z,RN),
respectively.For anyu∈E, let(2.5)φ(u)=1p∑n∈Z|Δu(n-1)|p-∑n∈Z[-K(n,u(n))+W(n,u(n))].To prove our results, we need the following generalization of Lebesgue's dominated convergence theorem.Lemma 2.1 (see [29]).
Let{fk(t)} and  {gk(t)} be two sequences of measurable functions on a measurable set  A, and let
(2.6)|fk(t)|≤gk(t),∀a.e.t∈A.
If
(2.7)lim⁡k→∞fk(t)=f(t),lim⁡k→∞gk(t)=g(t),∀a.e.t∈A,lim⁡k→∞∫Agk(t)dt=∫Ag(t)dt<+∞,
then
(2.8)lim⁡k→∞∫Afk(t)dt=∫Af(t)dt.Lemma 2.2.
Foru∈E,
(2.9)‖u‖∞≤‖u‖lp≤2‖u‖.Proof.
Sinceu∈E, it follows that lim|n|→∞|u(n)|=0. Hence, there exists n*∈ℤ such that
(2.10)‖u‖∞=|u(n*)|=max⁡n∈Z|u(n)|.
Hence, we have
(2.11)‖u‖∞≤(∑n∈Z|u(n)|p)1/p=‖u‖lp=(∑n∈Z|u(n)-u(n-1)+u(n-1)|p)1/p≤(∑n∈Z(|u(n)-u(n-1)|+|u(n-1)|)p)1/p≤(2p∑n∈Z(|u(n)-u(n-1)|p+|u(n-1)|p))1/p=2(∑n∈Z(|Δu(n-1)|p+|u(n-1)|p))1/p=2(∑n∈Z(|Δu(n-1)|p+|u(n)|p))1/p=2‖u‖.Lemma 2.3.
Suppose that  (K1), (K2),  and  (W2)  hold. Ifuk→u in  E, then  ∇K(n,uk)→∇K(n,u) and  ∇W(n,uk)→∇W(n,u) in  lp′(ℝ,ℝN), where  p′>1 satisfies  1/p+1/p′=1.Proof.
From (K1) and (K2), we have(2.12)|∇K(n,x)|≤pb2|x|p-1,∀(n,x)∈Z×RN.
Hence, from (2.12), we have
(2.13)|∇K(n,uk(n))-∇K(n,u(n))|p′≤[pb2(|uk(n)|p-1+|u(n)|p-1)]p′≤[pb22p-1|uk(n)-u(n)|p-1+pb2(1+2p-1)|u(n)|p-1]p′≤2pp′(pb2)p′|uk(n)-u(n)|p+2p′(pb2)p′(1+2p-1)p′|u(n)|p:=gk(n).
Moreover, since  uk→u in lp(ℤ,ℝN) and uk(n)→u(n) for almost every n∈ℤ, hence,
(2.14)lim⁡k→∞gk(n)=2p′(pb2)p′(1+2p-1)p′|u(n)|p:=g(n),∀a.e.n∈Z,lim⁡k→∞∑n∈Zgk(n)=lim⁡k→∞∑n∈Z[2pp′(pb2)p′|uk(n)-u(n)|p+2p′(pb2)p′(1+2p-1)p′|u(n)|p]=2pp′(pb2)p′lim⁡k→∞∑n∈Z|uk(n)-u(n)|p+2p′(pb2)p′(1+2p-1)p′∑n∈Z|u(n)|p=2p′(pb2)p′(1+2p-1)p′∑n∈Z|u(n)|p=∑n∈Zg(n)<+∞.
It follows from Lemma  2.1, (2.13), and the previous equations that
(2.15)lim⁡k→∞∑n∈Z|∇K(n,uk(n))-∇K(n,u(n))|p′=0.
This shows that ∇K(n,uk)→∇K(n,u) in lp′(ℤ,ℝN). By a similar proof, we can prove that ∇W(n,uk)→∇W(n,u) in lp′(ℤ,ℝN). The proof is complete.Lemma 2.4.
Under the conditions of Theorem1.1, one has
(2.16)〈φ′(u),v〉=∑n∈Z[|Δu(n-1)|p-2(Δu(n-1),Δv(n-1))+(∇K(n,u(n))-∇W(n,u(n)),v(n))|Δu(n-1)|p-2]
for u,v∈E, which yields that
(2.17)〈φ′(u),u〉=∑n∈Z[|Δu(n-1)|p+(∇K(n,u(n)),u(n))-(∇W(n,u(n)),u(n))].
Moreover, φ is continuously Fréchet-differential defined on E; that is,  φ∈C1(E,ℝ) and any critical point u of φ on E is classical solution of (1.1) with u(±∞)=0.Proof.
Firstly, we show thatφ:E→ℝ. Let u∈E, by (2.9) and (K1), we have
(2.18)∑n∈ZK(n,u(n))≤∑n∈Zb2|u(n)|p≤b22p‖u‖p.
By (W2), we get
(2.19)|W(n,x)|=|∫01(∇W(n,sx),x)ds|≤R|x|p,∀(n,x)∈Z×RN.
Hence, from (2.9) and (2.19), we have
(2.20)|∑n∈ZW(n,u(n))|≤∑n∈Z|W(n,u(n))|≤∑n∈ZR|u(n)|p≤R2p‖u‖p.
It follows from (2.5), (2.18), and (2.20) that φ:E→ℝ. Next we prove that φ∈C1(E,ℝ). Rewrite φ as follows:
(2.21)φ(u)=φ1(u)+φ2(u)-φ3(u),
where
(2.22)φ1(u):=1p∑n∈Z|Δu(n-1)|p,φ2(u):=∑n∈ZK(n,u(n)),φ3(u):=∑n∈ZW(n,u(n)).
It is easy to check that φ1∈C1(E,ℝ) and
(2.23)〈φ1′(u),v〉=∑n∈Z|Δu(n-1)|p-2(Δu(n-1),Δv(n-1)),∀u,v∈E.
Next, we prove that φi∈C1(E,ℝ),i=2,3, and
(2.24)〈φ2′(u),v〉=∑n∈Z(∇K(n,u(n)),v(n)),∀u,v∈E,(2.25)〈φ3′(u),v〉=∑n∈Z(∇W(n,u(n)),v(n)),∀u,v∈E.
For any u,v∈E and for any function θ:ℝ→(0,1), by (K2), we have
(2.26)∑n∈Zmax⁡h∈[0,1]|(∇K(n,u(n)+θ(t)hv(n)),v(n))|≤pb2∑n∈Zmax⁡h∈[0,1]|u(n)+θ(t)hv(n)|p-1|v(n)|≤2p-1pb2∑n∈Z(|u(n)|p-1+|v(n)|p-1)|v(n)|≤2p-1pb2[‖u‖lpp-1‖v‖lp+‖v‖lpp]<+∞.
Then by the previous equations and Lebesgue's dominated convergence theorem, we have
(2.27)〈φ2′(u),v〉=lim⁡h→0+φ2(u+hv)-φ2(u)h=lim⁡h→0+1h∑n∈Z[K(n,u(n)+hv(n))-K(n,u(n))]=lim⁡h→0+∑n∈Z(∇K(t,u(n)+θ(t)hv(n)),v(n))=∑n∈Z(∇K(n,u(n)),v(n)),∀u,v∈E.
Similarly, we can prove that (2.25) holds by using (W2) instead of (K2). Finally, we prove that φi∈C1(E,ℝ),i=2,3. Let uk→u in E; then by Lemma 2.3, we have
(2.28)|〈φ2′(uk)-φ2′(u),v〉|=|∑n∈Z(∇K(n,uk(n))-∇K(n,u(n)),v(n))|≤∑n∈Z|∇K(n,uk(n))-∇K(n,u(n))‖v(n)|≤‖v‖[∑n∈Z|∇K(n,uk(n))-∇K(n,u(n))|p′]1/p′⟶0,k⟶∞,∀v∈E.
This shows that φ2∈C1(E,ℝ). Similarly, we can prove that φ3∈C1(E,ℝ). Furthermore, by a standard argument, it is easy to show that the critical points of φ in E are classical solutions of (1.1) with u(±∞)=0. The proof is complete.Lemma 2.5 (see [30]).
LetE be a real Banach space with its dual space  E* and suppose that  φ∈C1(E,ℝ) satisfies
(2.29)max⁡{φ(0),φ(e)}≤η0<η≤inf⁡‖u‖=ρφ(u),
for some  η0<η, ρ>0, and e∈E with ∥e∥>ρ. Let c≥η be characterized by
(2.30)c=inf⁡Υ∈Γmax⁡0≤τ≤1φ(Υ(τ)),
where Γ={Υ∈C([0,1],E):Υ(0)=0,Υ(1)=e} is the set of continuous paths joining 0 to e; then there exists {uk}k∈ℕ⊂E such that
(2.31)φ(uk)⟶c,(1+‖uk‖)‖φ′(uk)‖E*⟶0ask⟶∞.


## 3. Proof of Theorem1.1

Proof of Theorem1.1.
We divide the proof of Theorem1.1 into three steps.
Step 1.
From (W1), there existsρ0>0 such that
(3.1)∇W(n,x)≤C12p|x|p-1,∀n∈Z,|x|≤ρ0,
where C1=min{1/p,b1}. From (3.1), we have
(3.2)W(n,x)=∫01(∇W(n,sx),x)ds≤∫01C12p|x|psp-1ds=C1p2p|x|p,∀n∈Z,|x|≤ρ0.
Let ρ=ρ0/2 and S={u∈E∣∥u∥=ρ}; then from (2.9), we obtain
(3.3)‖u‖∞≤ρ0,‖u‖lp≤2ρ,∀u∈S,
which together with (2.9), (3.2), and (K1) implies that
(3.4)φ(u)=1p∑n∈Z|Δu(n-1)|p-∑n∈Z[-K(n,u(n))+W(n,u(n))]≥1p∑n∈Z|Δu(n-1)|p+b1∑n∈Z|u(n)|p-∑n∈ZC1p2p|u(n)|p≥min⁡{1p,b1}‖u‖p-C1p2p‖u‖lpp≥C1‖u‖p-C1p‖u‖p=(p-1)C1p‖u‖p=α1>0,u∈S.Step 2.
From (K1), we have(3.5)φ(u)=1p∑n∈Z|Δu(n-1)|p-∑n∈Z[-K(n,u(n))+W(n,u(n))]≤1p∑n∈Z|Δu(n-1)|p+b2∑n∈Z|u(n)|p-∑n∈ZW(n,u(n))≤max⁡{1p,b2}‖u‖p-∑n∈ZW(n,u(n))≡C2‖u‖p-∑n∈ZW(n,u(n)).
By (W2) and (W3), we get
(3.6)lim⁡|x|→∞pW(n,x)|x|p=V∞(n)uniformlyforn∈Z.
Let W¯(n,x)=pW(n,x)-V∞(n)|x|p; it follows from (W2), (W3), (2.19), and (3.6) that
(3.7)W¯(n,x)≤(pR+sup⁡ZV∞(n))|x|p,∀x∈RN,lim⁡|x|→∞W¯(n,x)|x|p=0.
Define E1:={u(n)=xe-|n|:x∈ℝN,n∈ℤ}⊂E with
(3.8)inf⁡ZV∞(n)>max⁡⁡{1,pb2}(1+|1-e|n|-|n-1||p).
By an easy calculation, we have
(3.9)‖u‖p=(1+|1-e|n|-|n-1||p)‖u‖lpp.
In what follows, we prove that for some u∈E1 with ∥u∥=1, φ(su)→-∞ as s→∞. Otherwise, there exist a sequence {sk} with sk→∞ as k→∞ and a positive constant C3 such that φ(sku)≥-C3 for all k. From (3.5), we obtain
(3.10)-C3skp≤φ(sku)skp≤C2-1p∑n∈ZW¯(n,sku(n))skp-1p∑n∈ZV∞(n)|u(n)|p≤C2-1p∑n∈ZW¯(n,sku(n))skp-1pinf⁡ZV∞(n)‖u‖lpp.
It follows from (3.7) that
(3.11)W¯(n,sku(n))skp≤(pR+sup⁡ZV∞(n))|u(n)|p,W¯(n,sku(n))|sk|p⟶0ask⟶∞.
Hence, from Lebesgue's dominated theorem and (3.11), we have
(3.12)∑n∈ZW¯(n,snu(n))|sk|p⟶0ask⟶∞.
It follows from (3.8), (3.9), (3.10), and (3.12) that
(3.13)0⟵-C3skp≤C2-1p(1+|1-e|n|-|n-1||p)inf⁡ZV∞(n)<0ask⟶∞,
which is a contradiction. Hence, there exists e∈E with ∥e∥>ρ such that φ(e)≤0.Step 3.
From Step1, Step 2, and Lemma 2.5, we know that there is a sequence {uk}k∈ℕ⊂E such that
(3.14)φ(uk)⟶c,(1+‖uk‖)‖φ′(uk)‖E*⟶0ask⟶∞,
where E* is the dual space of E. In the following, we will prove that {uk}k∈ℕ is bounded in E. Otherwise, assume that ∥uk∥→∞ as k→∞. Let zk=uk/∥uk∥; we have ∥zk∥=1. It follows from (2.5), (2.16), (3.14), and (K2) that
(3.15)C4≥pφ(uk)-〈φ′(uk),uk〉=∑n∈Z[(∇W(n,uk(n)),uk(n))-pW(n,uk(n))]+∑n∈Z[pK(n,uk(n))-(∇K(n,uk(n)),uk(n))]≥∑n∈Z[(∇W(n,uk(n)),uk(n))-pW(t,uk(n))]:=∑n∈ZW̃(n,uk(n)).
Set Ωk(α,β)={n∈ℤ:α≤|uk(n)|≤β} for 0<α<β. Then from (3.15), we have
(3.16)C4≥∑n∈Ωk(0,α)W̃(n,uk(n))+∑n∈Ωk(α,β)W̃(n,uk(n))+∑n∈Ωk(β,+∞)W̃(n,uk(n)).
From (K1), (K2), and (3.14), we get
(3.17)o(1)=〈φ′(uk),uk〉=∑n∈Z[|Δuk(n-1)|p+(∇K(n,uk(n))-∇W(n,uk(n)),uk(n))]≥∑n∈Z[|Δuk(n-1)|p+b3|uk(n)|p-(∇W(n,uk(n)),uk(n))]≥min⁡{1,b3}‖uk‖p-∑n∈Z(∇W(n,uk(n)),uk(n)):=C5‖uk‖p-∑n∈Z(∇W(n,uk(n)),uk(n))=‖uk‖p(C5-∑n∈Z(∇W(n,uk(n)),uk(n))‖uk‖p),
which implies that
(3.18)lim⁡⁡sup⁡k→∞∑n∈Z(∇W(n,uk(n)),uk(n))|uk(n)|p|zk(n)|p=lim⁡⁡sup⁡k→∞∑n∈Z(∇W(n,uk(n)),uk(n))‖uk‖p≥C5.
Let 0<ɛ<C5/3. From (W1), there exists αɛ>0 such that
(3.19)|∇W(n,x)|≤ɛ2p|x|p-1for|x|≤αɛuniformlyforn∈Z.
Since ∥zk∥=1, it follows from (2.9) and (3.19) that
(3.20)∑n∈Ωk(0,αɛ)|∇W(n,uk(n))||uk(n)|p-1|zk(n)|p≤∑n∈Ωk(0,αɛ)ɛ2p|zk(n)|p≤ɛ,∀k∈N.
For s>0, let
(3.21)h(s):=inf⁡{W̃(n,x)∣n∈Z,x∈RNwith|x|≥s}.
Thus, from (W4), we have h(s)→+∞ as s→+∞, which together with (3.16) implies that
(3.22)meas⁡(Ωk(β,+∞))≤C6h(β)⟶0,asβ⟶+∞.
Hence, we can take βɛ sufficiently large such that
(3.23)∑n∈Ωk(βɛ,+∞)|zk(n)|p<ɛR.
The previous inequality and (W2) imply that
(3.24)∑n∈Ωk(βɛ,+∞)|∇W(n,uk(n))||uk(n)|p-1|zk(n)|p≤R∑n∈Ωk(βɛ,+∞)|zk(n)|p<ɛ,∀k∈N.
Next, for the previous 0<αɛ<βɛ, let
(3.25)cɛ:=inf⁡{W̃(n,x)|x|p:n∈Z,x∈RNwithαɛ≤|x|≤βɛ},dɛ:=max⁡{|∇W(n,x)||x|p-1:n∈Z,x∈RNwithαɛ≤|x|≤βɛ}.
From (W4), we have cɛ>0 and
(3.26)W̃(n,uk(n))≥cɛ|uk(n)|p,∀n∈Ωk(αɛ,βɛ).
From (3.15) and (3.26), we get
(3.27)∑n∈Ωk(αɛ,βɛ)|zk(n)|p=1‖uk‖p∑n∈Ωk(αɛ,βɛ)|uk(n)|p≤1‖uk‖p∑n∈Ωk(αɛ,βɛ)1cɛW̃(n,uk(n))≤C4cɛ‖uk‖p⟶0ask⟶∞,
which implies that
(3.28)∑n∈Ωk(αɛ,βɛ)|∇W(n,uk(n))||uk(n)|p-1|zk(n)|p≤dɛ∑n∈Ωk(αɛ,βɛ)|zk(n)|p⟶0ask⟶∞.
Therefore, there exists k0>0 such that
(3.29)∑n∈Ωk(αɛ,βɛ)|∇W(n,uk(n))||uk(n)|p-1|zk(n)|p≤ɛ,∀k≥k0.
It follows from (3.20), (3.24), and (3.29) that
(3.30)∑n∈Z(∇W(n,uk(n)),uk(n))|uk(n)|p|zk(n)|p≤∑n∈Z|∇W(t,uk(n))||uk(n)|p-1|zk(n)|p<3ɛ<C5,∀k≥k0,
which implies that
(3.31)lim⁡⁡sup⁡n→∞∑n∈Z(∇W(n,uk(n)),uk(n))|uk(n)|p|zk(n)|p<C5,
but this contradicts to (3.18). Hence, ∥uk∥ is bounded in E.
Going to a subsequence if necessary, we may assume that there existsu∈E such that uk⇀u as k→∞. In order to prove our theorem, it is sufficient to show that φ′(u)=0. For any a∈ℤ with a>0, let χa(t)=1 for t∈ℤ[-a,a] and let χa(t)=0 for t∈ℤ(-∞,-a)∪ℤ(a,∞). Then from (2.16), we have
(3.32)〈φ′(uk)-φ′(u),χa(uk-u)〉=∑n∈Z[-a,a]|Δuk(n-1)|p-2(Δuk(n-1),Δuk(n-1)-Δu(n-1))-∑n∈Z[-a,a]|Δu(n-1)|p-2(Δu(n-1),Δuk(n-1)-Δu(n-1))+∑n∈Z[-a,a](∇K(n,uk(n))-∇K(n,u(n)),uk(n)-u(n))-∑n∈Z[-a,a](∇W(n,uk(n))-∇W(n,u(n)),uk(n)-u(n))≥‖Δuk‖lpZ[-a,a]p+‖Δu‖lpZ[-a,a]p-∑n∈Z[-a,a]|Δuk(n-1)|p-1|Δu(n-1)|-∑n∈Z[-a,a]|Δu(n-1)|p-1|Δuk(n-1)|+∑n∈Z[-a,a](∇K(n,uk(n))-∇K(n,u(n)),uk(n)-u(n))-∑n∈Z[-a,a](∇W(n,uk(n))-∇W(n,u(n)),uk(n)-u(n))≥∥Δuk∥lpZ[-a,a]p+∥Δu∥lpZ[-a,a]p-∥Δu∥lpZ[-a,a]∥Δuk∥lpZ[-a,a]p-1-∥Δuk∥lpZ[-a,a]∥Δu∥lpZ[-a,a]p-1+∑n∈Z[-a,a](∇K(n,uk(n))-∇K(n,u(n)),uk(n)-u(n))-∑n∈Z[-a,a](∇W(n,uk(n))-∇W(n,u(n)),uk(n)-u(n))=(‖Δuk‖lpZ[-a,a]p-1-‖Δu‖lpZ[-a,a]p-1)(‖Δuk‖lpZ[-a,a]-‖Δu‖lpZ[-a,a])+∑n∈Z[-a,a](∇K(n,uk(n))-∇K(n,u(n)),uk(n)-u(n))-∑n∈Z[-a,a](∇W(n,uk(n))-∇W(n,u(n)),uk(n)-u(n)).
Since φ′(uk)→0 as k→+∞ and uk⇀u in E, it follows from (3.14) that
(3.33)〈φ′(uk)-φ′(u),χa(uk-u)〉⟶0ask⟶∞,∑n∈Z[-a,a](∇K(n,uk(n))-∇K(n,u(n)),uk(n)-u(n))⟶0ask⟶∞,∑n∈Z[-a,a](∇W(n,uk(n))-∇W(n,u(n)),uk(n)-u(n))⟶0ask⟶∞.
It follows from (3.32) and (3.33) that ∥Δuk∥lpℤ[-a,a]→∥Δu∥lpℤ[-a,a] as k→+∞.
For anyw∈C0∞(ℝ,ℝN), and assume that for some A∈ℤ with A>0, supp(w)⊂ℤ[-A,A]. Since
(3.34)lim⁡k→∞Δuk(n-1)=Δu(n-1),∀a.e.n∈Z,|(|Δuk(n-1)|p-2Δuk(n-1),Δw(n-1))|≤p-1p|Δuk(n-1)|p+1p|Δw(n-1)|p,∀n∈Z,k=1,2,…,lim⁡k→∞∑n∈Z[-A,A][p-1p|Δuk(n-1)|p+1p|Δw(n-1)|p]=p-1plim⁡k→∞‖Δuk‖lpZ[-A,A]p+1p‖Δw‖lpZ[-A,A]p=p-1p‖Δu‖lpZ[-A,A]p+1p‖Δw‖lpZ[-A,A]p=∑n∈Z[-A,A][p-1p|Δu(n-1)|p+1p|Δw(n-1)|p]<+∞,
then, we have
(3.35)∑n∈Z[-A,A](|Δuk(n-1)|p-2Δuk(n-1),Δw(n-1))⟶∑n∈Z[-A,A](|Δu(n-1)|p-2Δu(n-1),Δw(n-1))
as k→∞. Noting that
(3.36)∑n∈Z[-A,A](∇K(n,uk(n)),w(n))⟶∑n∈Z[-A,A](∇K(n,u(n)),w(n))ask⟶∞,∑n∈Z[-A,A](∇W(n,uk(n)),w(n))⟶∑n∈Z[-A,A](∇W(n,u(n)),w(n))ask⟶∞.
Hence, we have
(3.37)〈φ′(u),w〉=lim⁡k→∞〈φ′(uk),w〉=0,
which implies that φ′(u)=0; that is, u is a critical point of φ. From (K1) and (W1), we know that u≠0. In fact, if u=0, we have from (2.5), (K1), and (W1) that φ(u)=0. On the other hand, from Step 1, Step 2, and Lemma 2.5, we know that φ(u)=c>0. This is a contradiction. The proof of Theorem 1.1 is complete.


## 4. An Example

Example 4.1.
In problem (1.1), let p=3/2, and
(4.1)K(n,x)=(1+1|x|3/2+1)|x|3/2,W(n,x)=a(n)|x|3/2(1-1(ln⁡(e+|x|))1/2),
where a∈l∞(ℤ,ℝ+) with infℤa(n)>3. One can easily check that K satisfies conditions (K1) and (K2) with b1=1, b2=2, and b3=3/2. An easy computation shows that
(4.2)∇W(n,x)=32a(n)|x|-1/2x(1-1(ln⁡(e+|x|))1/2)+a(n)|x|1/2x2(e+|x|)(ln⁡(e+|x|))3/2,(∇W(n,x),x)-32W(n,x)=a(n)|x|5/22(e+|x|)(ln⁡(e+|x|))3/2.
Then it is easy to check that W satisfies (W1)–(W4). Hence, K(n,x) and W(n,x) satisfy all the conditions of Theorem 1.1 and then problem (1.1) has at least one nontrivial homoclinic solution.



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*Source: 351562-2011-08-10.xml*