Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 351562, 17 pages
doi:10.1155/2011/351562
Research Article
Existence of Homoclinic Orbits for
a Class of Asymptotically p-Linear Difference
Systems with p-Laplacian
Qiongfen Zhang and X. H. Tang
School of Mathematical Sciences and Computing Technology, Central South University, Changsha,
Hunan 410083, China
Correspondence should be addressed to X. H. Tang, tangxh@csu.edu.cn
Received 30 March 2011; Accepted 2 June 2011
Academic Editor: Yong Zhou
Copyright q 2011 Q. Zhang and X. H. Tang. This is an open access article distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
By applying a variant version of Mountain Pass Theorem in critical point theory, we prove the
existence of homoclinic solutions for the following asymptotically p-linear difference system with
p-Laplacian Δ|Δun − 1|p−2 Δun − 1 ∇−Kn, un Wn, un 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.
1. Introduction
Consider the following p-Laplacian difference system:
Δ |Δun − 1|p−2 Δun − 1 ∇−Kn, un Wn, un 0,
n ∈ Z,
1.1
where Δ is the forward difference operator defined by Δun un 1 − un, Δ2 un ΔΔun, 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 un of 1.1 is homoclinic to 0 if un → 0 as n → ±∞. In
addition, if un /
≡ 0, then un is called a nontrivial homoclinic solution.
When p 2, 1.1 can be regarded as a discrete analogue of the following second-order
Hamiltonian system:
üt ∇−Kt, ut Wt, ut 0,
t ∈ R.
1.2
2
Abstract and Applied Analysis
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 1th
generation xn 1 is a function of the nth generation xn. 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 that K and W satisfy the following conditions.
K1 There are two positive constants b1 and b2 such that
b1 |x|p ≤ Kn, x ≤ b2 |x|p ,
∀n, x ∈ Z × RN .
1.3
K2 There is a positive constant b3 such that
b3 |x|p ≤ ∇Kn, x, x ≤ |∇Kn, x||x| ≤ pKn, x,
∀n, x ∈ Z × RN .
1.4
W1 Wn, 0 0, ∇Wn, x o|x|p−1 as |x| → 0 uniformly for n ∈ Z.
W2 There exists a constant R > 0 such that
|∇Wn, x|
|x|p−1
≤ R,
∀n ∈ Z, x ∈ RN .
1.5
W3 There exists a function V∞ ∈ l∞ Z, R such that
lim
|x| → ∞
∇Wn, x − V∞ n|x|p−2 x
|x|p−1
0
uniformly for n ∈ Z,
infV∞ n > max 1, pb2 .
Z
1.6
Abstract and Applied Analysis
3
x ∇Wn, x, x − pWn, x,
W4 Wn,
x ∞
lim Wn,
uniformly for n ∈ Z,
|x| → ∞
1.7
and for any fixed 0 < c1 < c2 < ∞,
x
Wn,
> 0.
n∈Z, c1 ≤|x|≤c2
|x|p
1.8
inf
Then problem 1.1 has at least one nontrivial homoclinic solution.
Remark 1.2. The function Wn, x in this paper is asymptotically p-linear at infinity. The
behavior of the gradient of Wn, x at infinity is like that of a function V∞ n|x|p−2 x, 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 Wn, x at infinity
cannot be found in the literature. From this point, our result is new.
2. Preliminaries
Let
S {un}n∈Z : un ∈ RN , n ∈ Z ,
E
u∈S:
p
p
|Δun − 1| |un|
2.1
< ∞ ,
n∈Z
and for u ∈ E, let
u |Δun − 1|p |un|p < ∞
1/p
2.2
.
n∈Z
Then E is a uniform convex Banach space with this norm. As usual, for 1 ≤ p < ∞, let
l
p
Z, R
N
u∈S:
p
|un| < ∞ ,
l
∞
Z, R
N
u ∈ S : sup|un| < ∞ , 2.3
n∈Z
n∈Z
and their norms are given by
ulp 1/p
p
|un|
,
∀u ∈ lp Z, RN ,
n∈Z
u∞ sup{|un| : n ∈ Z},
respectively.
∀u ∈ l
∞
Z, RN ,
2.4
4
Abstract and Applied Analysis
For any u ∈ E, let
ϕu 1
−Kn, un Wn, un.
|Δun − 1|p −
p n∈Z
n∈Z
2.5
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
fk t ≤ gk t,
∀a.e. t ∈ A.
2.6
If
lim fk t ft,
k→∞
lim gk t gt,
k→∞
k→∞
2.7
gtdt < ∞,
gk tdt lim
∀a.e. t ∈ A,
A
A
then
fk tdt lim
k→∞
A
2.8
ftdt.
A
Lemma 2.2. For u ∈ E,
u∞ ≤ ulp ≤ 2u.
2.9
Proof. Since u ∈ E, it follows that lim|n| → ∞ |un| 0. Hence, there exists n∗ ∈ Z such that
u∞ |un∗ | max|un|.
2.10
n∈Z
Hence, we have
u∞ ≤
1/p
|un|
p
ulp n∈Z
≤
1/p
|un − un − 1 un − 1|
n∈Z
1/p
|un − un − 1| |un − 1|
p
n∈Z
≤
p
2
1/p
p
p
|un − un − 1| |un − 1|
n∈Z
p
Abstract and Applied Analysis
5
1/p
p
|Δun − 1| |un − 1|
2
p
n∈Z
2
1/p
p
|Δun − 1| |un|
p
n∈Z
2u.
2.11
Lemma 2.3. Suppose that (K1), (K2), and (W2) hold. If uk → u in E, then ∇Kn, uk → ∇Kn, u
and ∇Wn, uk → ∇Wn, u in lp R, RN , where p > 1 satisfies 1/p 1/p 1.
Proof. From K1 and K2, we have
|∇Kn, x| ≤ pb2 |x|p−1 ,
2.12
∀n, x ∈ Z × RN .
Hence, from 2.12, we have
p
|∇Kn, uk n − ∇Kn, un|p ≤ pb2 |uk n|p−1 |un|p−1
p ≤ pb2 2p−1 |uk n − un|p−1 pb2 1 2p−1 |un|p−1
p p p 1 2p−1 |un|p
≤ 2pp pb2 |uk n − un|p 2p pb2
: gk n.
2.13
Moreover, since uk → u in lp Z, RN and uk n → un for almost every n ∈ Z, hence,
p p 1 2p−1 |un|p : gn,
lim gk n 2p pb2
k→∞
lim
k→∞
n∈Z
gk n lim
∀a.e. n ∈ Z,
p p p 1 2p−1 |un|p
2pp pb2 |uk n − un|p 2p pb2
k→∞
n∈Z
p p p 1 2p−1
2pp pb2 lim
|uk n − un|p 2p pb2
|un|p
k→∞
n∈Z
n∈Z
p p 1 2p−1
2p pb2
|un|p
n∈Z
n∈Z
gn < ∞.
2.14
6
Abstract and Applied Analysis
It follows from Lemma 2.1, 2.13, and the previous equations that
lim
k→∞
n∈Z
|∇Kn, uk n − ∇Kn, un|p 0.
2.15
This shows that ∇Kn, uk → ∇Kn, u in lp Z, RN . By a similar proof, we can prove that
∇Wn, uk → ∇Wn, u in lp Z, RN . The proof is complete.
Lemma 2.4. Under the conditions of Theorem 1.1, one has
ϕ u, v |Δun − 1|p−2 Δun − 1, Δvn − 1
n∈Z
∇Kn, un − ∇Wn, un, vn
2.16
for u, v ∈ E, which yields that
ϕ u, u |Δun − 1|p ∇Kn, un, un − ∇Wn, un, un .
n∈Z
2.17
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 that ϕ : E → R. Let u ∈ E, by 2.9 and K1, we have
Kn, un ≤
n∈Z
b2 |un|p ≤ b2 2p up .
n∈Z
2.18
By W2, we get
1
|Wn, x| ∇Wn, sx, xds ≤ R|x|p ,
0
∀n, x ∈ Z × RN .
2.19
Hence, from 2.9 and 2.19, we have
R|un|p ≤ R2p up .
|Wn, un| ≤
Wn, un ≤
n∈Z
n∈Z
n∈Z
2.20
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:
ϕu ϕ1 u ϕ2 u − ϕ3 u,
2.21
Abstract and Applied Analysis
7
where
ϕ1 u :
1
|Δun − 1|p ,
p n∈Z
ϕ2 u :
ϕ3 u :
Kn, un,
n∈Z
Wn, un.
2.22
n∈Z
It is easy to check that ϕ1 ∈ C1 E, R and
ϕ1 u, v |Δun − 1|p−2 Δun − 1, Δvn − 1,
∀u, v ∈ E.
n∈Z
2.23
Next, we prove that ϕi ∈ C1 E, R, i 2, 3, and
ϕ2 u, v ∇Kn, un, vn,
∀u, v ∈ E,
2.24
∀u, v ∈ E.
2.25
n∈Z
ϕ3 u, v ∇Wn, un, vn,
n∈Z
For any u, v ∈ E and for any function θ : R → 0, 1, by K2, we have
n∈Z
max |∇Kn, un θthvn, vn| ≤ pb2
h∈0,1
max |un θthvn|p−1 |vn|
h∈0,1
n∈Z
≤ 2p−1 pb2
|un|p−1 |vn|p−1 |vn|
n∈Z
p−1
p
≤ 2p−1 pb2 ulp vlp vlp
2.26
< ∞.
Then by the previous equations and Lebesgue’s dominated convergence theorem, we have
ϕ2 u hv − ϕ2 u
ϕ2 u, v lim
h→0
h
lim
h→0
lim
h→0
n∈Z
1
Kn, un hvn − Kn, un
h n∈Z
∇Kt, un θthvn, vn
n∈Z
∇Kn, un, vn,
∀u, v ∈ E.
2.27
8
Abstract and Applied Analysis
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
ϕ uk − ϕ u, v 2
2
∇Kn, uk n − ∇Kn, un, vn
n∈Z
≤
2.28
|∇Kn, uk n − ∇Kn, unvn|
n∈Z
≤ v
1/p
|∇Kn, uk n − ∇Kn, un|
p
−→ 0,
k −→ ∞, ∀v ∈ E.
n∈Z
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. Let E be a real Banach space with its dual space E∗ and suppose that ϕ ∈
C1 E, R satisfies
max ϕ0, ϕe ≤ η0 < η ≤ inf ϕu,
uρ
2.29
for some η0 < η, ρ > 0, and e ∈ E with e > ρ. Let c ≥ η be characterized by
c inf max ϕΥτ,
2.30
Υ∈Γ 0≤τ≤1
where Γ {Υ ∈ C0, 1, E : Υ0 0, Υ1 e} is the set of continuous paths joining 0 to e; then
there exists {uk }k∈N ⊂ E such that
ϕuk −→ c,
1 uk ϕ uk E∗ −→ 0
as k −→ ∞.
2.31
3. Proof of Theorem 1.1
Proof of Theorem 1.1. We divide the proof of Theorem 1.1 into three steps.
Step 1. From W1, there exists ρ0 > 0 such that
∇Wn, x ≤
C1 p−1
|x| ,
2p
∀n ∈ Z, |x| ≤ ρ0 ,
3.1
Abstract and Applied Analysis
9
where C1 min{1/p, b1 }. From 3.1, we have
Wn, x 1
∇Wn, sx, xds
0
≤
1
0
3.2
C1 p p−1
C1
|x| s ds p |x|p ,
2p
p2
∀n ∈ Z, |x| ≤ ρ0 .
Let ρ ρ0 /2 and S {u ∈ E | u ρ}; then from 2.9, we obtain
u∞ ≤ ρ0 ,
ulp ≤ 2ρ,
∀u ∈ S,
3.3
which together with 2.9, 3.2, and K1 implies that
ϕu 1
−Kn, un Wn, un
|Δun − 1|p −
p n∈Z
n∈Z
C1
1
|Δun − 1|p b1 |un|p −
|un|p
p
p n∈Z
p2
n∈Z
n∈Z
C1
1
p
, b1 up − p ulp
≥ min
p
p2
p − 1 C1
C1
≥ C1 up −
up up α1 > 0, u ∈ S.
p
p
≥
3.4
Step 2. From K1, we have
ϕu 1
−Kn, un Wn, un
|Δun − 1|p −
p n∈Z
n∈Z
1
Wn, un
|Δun − 1|p b2 |un|p −
p n∈Z
n∈Z
n∈Z
1
, b2 up −
Wn, un
≤ max
p
n∈Z
≡ C2 up −
Wn, un.
≤
3.5
n∈Z
By W2 and W3, we get
lim
|x| → ∞
pWn, x
V∞ n
|x|p
uniformly for n ∈ Z.
3.6
10
Abstract and Applied Analysis
Let Wn, x pWn, x − V∞ n|x|p ; it follows from W2, W3, 2.19, and 3.6 that
pR supV∞ n |x|p ,
Wn, x ≤
Z
∀x ∈ RN ,
lim
|x| → ∞
Wn, x
0.
|x|p
3.7
Define E1 : {un xe−|n| : x ∈ RN , n ∈ Z} ⊂ E with
p inf V∞ n > max 1, pb2 1 1 − e|n|−|n−1| .
3.8
Z
By an easy calculation, we have
p p
up 1 1 − e|n|−|n−1| ulp .
3.9
In what follows, we prove that for some u ∈ E1 with u 1, ϕsu → −∞ as s → ∞.
Otherwise, there exist a sequence {sk } with sk → ∞ as k → ∞ and a positive constant C3
such that ϕsk u ≥ −C3 for all k. From 3.5, we obtain
−
C3
≤
p
sk
ϕsk u
p
sk
≤ C2 −
1 Wn, sk un 1 −
V∞ n|un|p
p
p n∈Z
p
s
n∈Z
k
1 Wn, sk un 1
p
≤ C2 −
− inf V∞ nulp .
p
p n∈Z
p Z
s
3.10
k
It follows from 3.7 that
Wn, sk un
p
sk
≤
pR sup V ∞ n |un|p ,
Z
Wn, sk un
−→ 0
|sk |p
as k −→ ∞.
3.11
Hence, from Lebesgue’s dominated theorem and 3.11, we have
Wn, sn un
n∈Z
|sk |p
−→ 0
as k −→ ∞.
3.12
It follows from 3.8, 3.9, 3.10, and 3.12 that
0 ←− −
C3
p
sk
1
≤ C2 − p inf V∞ n < 0
Z
p 1 1 − e|n|−|n−1| as k −→ ∞,
which is a contradiction. Hence, there exists e ∈ E with e > ρ such that ϕe ≤ 0.
3.13
Abstract and Applied Analysis
11
Step 3. From Step 1, Step 2, and Lemma 2.5, we know that there is a sequence {uk }k∈N ⊂ E
such that
1 uk ϕ uk E∗ −→ 0
ϕuk −→ c,
as k −→ ∞,
3.14
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
C4 ≥ pϕuk − ϕ uk , uk
∇Wn, uk n, uk n − pWn, uk n
n∈Z
pKn, uk n − ∇Kn, uk n, uk n
3.15
n∈Z
uk n.
≥
Wn,
∇Wn, uk n, uk n − pWt, uk n :
n∈Z
n∈Z
Set Ωk α, β {n ∈ Z : α ≤ |uk n| ≤ β} for 0 < α < β. Then from 3.15, we have
C4 ≥
uk n Wn,
n∈Ωk 0,α
uk n Wn,
n∈Ωk α,β
uk n.
Wn,
3.16
n∈Ωk β,∞
From K1, K2, and 3.14, we get
o1 ϕ uk , uk
|Δuk n − 1|p ∇Kn, uk n − ∇Wn, uk n, uk n
n∈Z
≥
|Δuk n − 1|p b3 |uk n|p − ∇Wn, uk n, uk n
n∈Z
≥ min{1, b3 }uk p −
: C5 uk p −
uk C5 −
3.17
∇Wn, uk n, uk n
n∈Z
∇Wn, uk n, uk n
n∈Z
p
∇Wn, uk n, uk n
n∈Z
uk p
,
which implies that
lim sup
∇Wn, uk n, uk n
k → ∞ n∈Z
|uk n|
p
|zk n|p lim sup
∇Wn, uk n, uk n
k → ∞ n∈Z
uk p
≥ C5 .
3.18
12
Abstract and Applied Analysis
Let 0 < ε < C5 /3. From W1, there exists αε > 0 such that
|∇Wn, x| ≤
ε p−1
|x|
2p
for |x| ≤ αε uniformly for n ∈ Z.
3.19
Since zk 1, it follows from 2.9 and 3.19 that
|∇Wn, uk n|
|uk n|
n∈Ωk 0,αε p−1
|zk n|p ≤
n∈Ωk
ε
|z n|p ≤ ε,
p k
2
0,α ∀k ∈ N.
3.20
ε
For s > 0, let
x | n ∈ Z, x ∈ RN with |x| ≥ s .
hs : inf Wn,
3.21
Thus, from W4, we have hs → ∞ as s → ∞, which together with 3.16 implies that
C6
meas Ωk β, ∞ ≤ −→ 0,
h β
as β −→ ∞.
3.22
Hence, we can take βε sufficiently large such that
|zk n|p <
n∈Ωk βε ,∞
ε
.
R
3.23
The previous inequality and W2 imply that
n∈Ωk βε ,∞
|∇Wn, uk n|
|uk n|
p−1
|zk n|p ≤ R
|zk n|p < ε,
∀k ∈ N.
n∈Ωk βε ,∞
3.24
Next, for the previous 0 < αε < βε , let
x
Wn,
N
cε : inf
: n ∈ Z, x ∈ R with αε ≤ |x| ≤ βε ,
|x|p
|∇Wn, x|
N
dε : max
: n ∈ Z, x ∈ R with αε ≤ |x| ≤ βε .
|x|p−1
3.25
From W4, we have cε > 0 and
uk n ≥ cε |uk n|p ,
Wn,
∀n ∈ Ωk αε , βε .
3.26
Abstract and Applied Analysis
13
From 3.15 and 3.26, we get
|zk n|p n∈Ωk αε ,βε 1
|uk n|p
p
uk n∈Ωk αε ,βε ≤
1
1
Wn, uk n
p
uk n∈Ω α ,β cε
k ε ε
≤
C4
−→ 0
cε uk p
3.27
as k −→ ∞,
which implies that
|∇Wn, uk n|
|uk n|
n∈Ωk αε ,βε p−1
|zk n|p ≤ dε
|zk n|p −→ 0
as k −→ ∞.
n∈Ωk αε ,βε 3.28
Therefore, there exists k0 > 0 such that
|∇Wn, uk n|
n∈Ωk αε ,βε |uk n|
p−1
|zk n|p ≤ ε,
∀k ≥ k0 .
3.29
It follows from 3.20, 3.24, and 3.29 that
∇Wn, uk n, uk n
|uk n|
n∈Z
p
|zk n|p ≤
|∇Wt, uk n|
n∈Z
|uk n|
p−1
|zk n|p < 3ε < C5 ,
∀k ≥ k0 , 3.30
which implies that
lim sup
∇Wn, uk n, uk n
n → ∞ n∈Z
|uk n|p
|zk n|p < C5 ,
3.31
but this contradicts to 3.18. Hence, uk is bounded in E.
Going to a subsequence if necessary, we may assume that there exists u ∈ 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∪Za, ∞.
Then from 2.16, we have
ϕ uk − ϕ u, χa uk − u
|Δuk n − 1|p−2 Δuk n − 1, Δuk n − 1 − Δun − 1
n∈Z−a,a
−
|Δun − 1|p−2 Δun − 1, Δuk n − 1 − Δun − 1
n∈Z−a,a
∇Kn, uk n − ∇Kn, un, uk n − un
n∈Z−a,a
14
Abstract and Applied Analysis
−
∇Wn, uk n − ∇Wn, un, uk n − un
n∈Z−a,a
p
p
≥ Δuk lp Z−a,a Δulp Z−a,a −
−
|Δuk n − 1|p−1 |Δun − 1|
n∈Z−a,a
|Δun − 1|p−1 |Δuk n − 1|
n∈Z−a,a
∇Kn, uk n − ∇Kn, un, uk n − un
n∈Z−a,a
−
∇Wn, uk n − ∇Wn, un, uk n − un
n∈Z−a,a
p
p
p−1
p−1
≥ Δuk lp Z−a,a Δulp Z−a,a − Δulp Z−a,a Δuk lp Z−a,a − Δuk lp Z−a,a Δulp Z−a,a
∇Kn, uk n − ∇Kn, un, uk n − un
n∈Z−a,a
−
∇Wn, uk n − ∇Wn, un, uk n − un
n∈Z−a,a
p−1
p−1
Δuk lp Z−a,a − Δulp Z−a,a Δuk lp Z−a,a − Δulp Z−a,a
∇Kn, uk n − ∇Kn, un, uk n − un
n∈Z−a,a
−
∇Wn, uk n − ∇Wn, un, uk n − un.
n∈Z−a,a
3.32
Since ϕ uk → 0 as k → ∞ and uk u in E, it follows from 3.14 that
ϕ uk − ϕ u, χa uk − u −→ 0
as k −→ ∞,
∇Kn, uk n − ∇Kn, un, uk n − un −→ 0
as k −→ ∞,
n∈Z−a,a
∇Wn, uk n − ∇Wn, un, uk n − un −→ 0
3.33
as k −→ ∞.
n∈Z−a,a
It follows from 3.32 and 3.33 that Δuk lp Z−a,a → Δulp Z−a,a as k → ∞.
For any w ∈ C0∞ R, RN , and assume that for some A ∈ Z with A > 0, suppw ⊂
Z−A, A. Since
lim Δuk n − 1 Δun − 1,
k→∞
∀a.e. n ∈ Z,
|Δuk n − 1|p−2 Δuk n − 1, Δwn − 1 Abstract and Applied Analysis
15
p−1
1
|Δuk n − 1|p |Δwn − 1|p , ∀n ∈ Z, k 1, 2, . . . ,
p
p
p−1
1
lim
|Δuk n − 1|p |Δwn − 1|p
k→∞
p
p
n∈Z−A,A
≤
p−1
1
p
p
lim Δuk lp Z−A,A Δwlp Z−A,A
p k→∞
p
p−1
1
p
p
Δulp Z−A,A Δwlp Z−A,A
p
p
p − 1
1
p
p
|Δun − 1| |Δwn − 1| < ∞,
p
p
n∈Z−A,A
3.34
then, we have
|Δuk n − 1|p−2 Δuk n − 1, Δwn − 1
n∈Z−A,A
−→
|Δun − 1|p−2 Δun − 1, Δwn − 1
3.35
n∈Z−A,A
as k → ∞. Noting that
∇Kn, uk n, wn −→
n∈Z−A,A
∇Kn, un, wn
as k −→ ∞,
n∈Z−A,A
∇Wn, uk n, wn −→
n∈Z−A,A
3.36
∇Wn, un, wn
as k −→ ∞.
n∈Z−A,A
Hence, we have
ϕ u, w lim ϕ uk , w 0,
3.37
k→∞
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 ϕu 0. On the other
hand, from Step 1, Step 2, and Lemma 2.5, we know that ϕu 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
Kn, x 1
1
|x|3/2 1
|x|
3/2
,
Wn, x an|x|
3/2
1−
1
lne |x|1/2
,
4.1
16
Abstract and Applied Analysis
where a ∈ l∞ Z, R with infZ an > 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
3
1
an|x|1/2 x
−1/2
∇Wn, x an|x|
x 1−
,
2
2e |x|lne |x|3/2
lne |x|1/2
an|x|5/2
3
.
∇Wn, x, x − Wn, x 2
2e |x|lne |x|3/2
4.2
Then it is easy to check that W satisfies W1–W4. Hence, Kn, x and Wn, x satisfy all
the conditions of Theorem 1.1 and then problem 1.1 has at least one nontrivial homoclinic
solution.
Acknowledgment
This work is partially supported by the NNSF no. 10771215 of China.
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