Volume 2010, Article ID 195376, 17 pagesdoi:10.1155/2010/195376 Research Article On Homoclinic Solutions of a Semilinear p-Laplacian Difference Equation with Periodic Coefficients Albert
Trang 1Volume 2010, Article ID 195376, 17 pages
doi:10.1155/2010/195376
Research Article
On Homoclinic Solutions of a Semilinear
p-Laplacian Difference Equation with Periodic
Coefficients
Alberto Cabada,1 Chengyue Li,2 and Stepan Tersian3
1 Departamento de An´alise Matem´atica, Facultade de Matem´aticas, Universidade de
Santiago de Compostela, 15782 Santiago de Compostela, Spain
2 Department of Mathematics, Minzu University of China, Beijing 100081, China
3 Department of Mathematical Analysis, University of Rousse, 7017 Rousse, Bulgaria
Correspondence should be addressed to Alberto Cabada,alberto.cabada@usc.es
Received 5 July 2010; Accepted 27 October 2010
Academic Editor: Jianshe Yu
Copyrightq 2010 Alberto Cabada et al 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
We study the existence of homoclinic solutions for semilinear p-Laplacian difference equations
with periodic coefficients The proof of the main result is based on Brezis-Nirenberg’s Mountain Pass Theorem Several examples and remarks are given
1 Introduction
This paper is concerned with the study of the existence of homoclinic solutions for the
p-Laplacian difference equation
Δ2
p u k − 1 − V kuk|uk| q−2 λfk, uk 0, ut → 0, |t| → ∞,
1.1
where uk, k ∈ is a sequence or real numbers, Δ is the difference operator Δuk uk
1 − uk,
Δ2
p u k − 1 Δuk|Δuk| p−2− Δuk − 1|Δuk − 1| p−2 1.2
Trang 2is referred to as the p-Laplacian difference operator, and functions V k and fk, x are T-periodic in k and satisfy suitable conditions.
In the theory of differential equations, a trajectory xt, which is asymptotic to a constant as|t| → ∞ is called doubly asymptotic or homoclinic orbit The notion of homoclinic
orbit is introduced by Poincar´e 1
Recently, there is a large literature on the use of variational methods to the existence of homoclinic or heteroclinic orbits of Hamiltonian systems; see 2 7
In the recent paper of Li 8
for some classes of ODE’s with periodic potentials is presented It is based on the Brezis and Nirenberg’s mountain-pass theorem 9
orbits for discrete p-Laplacian type equations.
Discrete boundary value problems have been intensively studied in the last decade The studies of such kind of problems can be placed at the interface of certain mathematical fields, such as nonlinear differential equations and numerical analysis On the other hand, they are strongly motivated by their applicability to mathematical physics and biology The variational approach to the study of various problems for difference equations has been recently applied in, among others, the papers of Agarwal et al 10 11
Along the paper, given two integer numbers a < b, we will denote
Moreover, for every p > 1, we consider the following function
ϕ p t t|t| p−2, Φp t |t| p
It is obvious thatΦ
p t ϕ p t for all t ∈and p / 0 Moreover
Δ2
Suppose that
0 < V0 min{V 0, , V T − 1} ≤ max{V 0, , V T − 1} V1. 1.6 Denote
A u
k∈
Φp Δuk − 1
k∈
Let us consider functions f satisfying the following assumptions.
F1 The function fk, t is continuous in t ∈ and T-periodic in k.
Trang 3F2 The potential function Fk, t of fk, t
F k, t
t
0
satisfies the Rabinowitz’s type condition:
There exist μ > p ≥ q > 1 and s > 0 such that
μF k, t ≤ tfk, t, k ∈ , t / 0,
F3 fk, t o|t| q−1 as |t| → 0.
Further we consider the semilinear eigenvalue p-Laplacian difference equation
Δ2
p u k − 1 − V kuk|uk| q−2 λfk, uk 0, 1.10
where λ > 0 and we are looking for its homoclinic solutions, that is, solutions of1.10 such
that uk → 0 as |k| → ∞.
In order to obtain homoclinic solutions of1.10, we will use variational approach and Brezis-Nirenberg mountain pass theorem 9
To this end, consider the functional J : q → , defined as
J u Au − λ
k∈
Our main result is the following
f k, · : × → satisfy assumptions F1–F3 Then, for each λ > 0, 1.10 has a nonzero
homoclinic solution u ∈ q , which is a critical point of the functional J : q → .
Moreover, given a nontrivial solution u of problem1.10, there exist k±two integer numbers such that for all k > kand k < k−, the sequence u k is strictly monotone.
The paper is organized as follows InSection 2, we present the proof of the main result and discuss the optimality of the condition F2 In Section 3, we give some examples of equations modeled by this kind of problems and present some additional remarks
Trang 42 Proof of the Main Result
Let u {uk : k ∈ } be a sequence, q > 1 and
q
u : |u| q
q
k∈
|uk| q <∞
,
∞
u : |u|∞ sup
k∈ |uk| < ∞
.
2.1
It is well known that if 0 < q ≤ p, then q ⊆ p Indeed, if
k∈ |uk| q <∞, there exists
a positive integer number R, such that for all k satisfying |k| > R it is verified that |uk| q < 1
and, as consequence,|uk| p ≤ |uk| q and the series
k∈ |uk| pis convergent too
Consider now the functional J : q → , defined as
J u Au − λ
k∈
with A given in1.7 and F defined in 1.8
We have the following result
Lemma 2.1 The functional J : q → is well defined, C1-differentiable, and its critical points are solutions of 1.10.
Proof By using the inequality for nonnegative a and b and p > 1
a b
2
p
≤ a p b p
and the inclusion q ⊆ p for 1 < q ≤ p, it follows that
k∈
|Δuk − 1| p≤ 2p−1
k∈
|uk| p |uk − 1| p
2p
k∈
|uk| p < ∞. 2.4
Now, let us see that the series
k∈ F k, uk is convergent: by using F3, it follows
that there exist δ ∈ 0, 1 and sufficiently large N such that
F k, uk < |uk| q for|uk| q < δ < 1, |k| > N. 2.5
Then, the series
k∈ F k, uk is convergent and the functional J is well defined on
q
Trang 5It is Gˆateaux differentiable and for v ∈ q:
Ju, v lim
t→ 0
J u tv − Ju
t
k∈
Δuk − 1|Δuk − 1| p−2Δvk − 1
k∈
V kuk|uk| q−2v k − λ
k∈
f k, ukvk
2.6
and partial derivatives
∂J u
∂u k −Δ2p u k − 1 V kuk|uk| q−2− λfk, uk, 2.7
are continuous functions
Moreover the functional J is continuously Fr´echet-differentiable in q It is clear, by
2.7, that the critical points of J are solutions of 1.10
To obtain homoclinic solutions of1.10 we will use mountain-pass theorem of Brezis
be a C1-functional I satisfies the PS c condition if every sequence x k of X such that
has a convergent subsequence A sequencex k ⊂ X such that 2.8 holds is referred to as
PS c -sequence.
norm 1X,
ii Ie < 0.
Let c infγ∈Γ{max0≤t≤1I γt} ≥ α, where
Then, there exists a PS c sequence for I Moreover, if I satisfies the PS c condition, then c is a critical value of I, that is, there exists u0∈ X such that Iu0 c and Iu0 0.
Note that, by assumption1.5, the norm | · |q in qis equivalent to
q
q 1
q
k∈
Trang 6Lemma 2.3 Suppose that F1–F3 hold, then there exist ρ > 0, α > 0 and e ∈ q such that q > ρ and
q ρ,
2 Je < 0.
Proof By F3, there exists δ ∈ 0, 1 such that
F k, t ≤ V0
Let ρ V0/q1/q δ V0defined in1.6 q ρ,
V0
q δ
q ρ q q
q 1
q
k∈
V k|uk| q
≥ V0
q |uk| q for all k ∈ ,
2.12
which implies that|uk| ≤ δ for all k ∈
Hence, by2.11
k∈
F k, uk ≤ V0
2qλ
k∈
|uk| q
2qλ
k∈
V k|uk| q 1
2λ
q q
J u Au − λ
k∈
F k, uk
q
q−1 2
q
q 1 2
q
q ρ q
2 > 0.
2.13
ByF2, there exist c1, c2 > 0 such that F k, t ≥ c1 t μ − c2for all t > 0 and k∈
Take v ∈ q , v 0 a > 0, vk 0 if k / 0 Then, since μ > p ≥ q
J κv Aκv − λ
k∈
F k, κvk
≤ 2
p κ
p a p V 0 κ q a q
q − λc1κμ a μ − c2
< 0,
2.14
if κ is sufficiently large.
Then, we can take κ large enough, such that for e q q V 0κ q a q /q > ρ qand
2.14 holds
Trang 7Lemma 2.4 Suppose that the assumptions of Lemma 2.3 hold Then, there exists c > 0 and a q -bounded PS c sequence for J.
Proof ByLemma 2.3andTheorem 2.2there exists a sequenceu m ⊂ qsuch that
where
c inf
γ∈Γ
max
t J
γ t,
Γ γ q : γ0 0, γ1 e,
2.16
and e is defined in the proof ofLemma 2.3
We will prove that the sequenceu m is bounded in q We have for μ > p ≥ q
Ju m , u m
k∈
|Δu m k − 1| p
k∈
V k|u m k| q − λ
k∈
f k, u m ku m k, 2.17
and, byF2,
μJ u m − Ju m , u m
μ
p − 1
k∈
|Δu m k − 1| p
μ
q − 1
k∈
V k|u m k| q
− λ
k∈
μF k, u m k − fk, u m ku m k
≥
μ
q − 1 q m q q μ − q m q q ,
2.18
which implies that the sequence u m is bounded in q
Now we are in a position to proveTheorem 1.1
Proof of Theorem 1.1 For any m∈ , the sequence{|u m k|, k ∈ Z}, given inLemma 2.4, is
bounded in qand, in consequence,|u m k| → 0 as |k| → ∞ Let |u m k| takes its maximum
at k m ∈ There exists a unique j m ∈ , such that j m T ≤ k m < j m 1T and let w m k
u m kj m T Then |w m k| takes its maximum at i m k m −j m T
of V and f ·, t, it follows that
m q m q ,
Trang 8Sinceu m is bounded in q , there exists w ∈ q , such that w m w weakly in q The weak
convergence in q implies that w m k → wk for every k ∈ Indeed, if we take a test function v k ∈ q , v k k 1, v k j 0 if j / k, then
Moreover, for any v ∈ q
Jw m , v Ju m , v· j m T
≤Ju m
∗v
· j m T
q
Ju m
∗ q −→ 0,
2.21
which implies that Jw m → 0, which means that for every v ∈ q,
k∈
ϕ p Δw m k − 1Δvk − 1
k∈
V kϕ q w m kvk
k∈
f k, w m kvk −→ 0, ∀k ∈ , as m −→ ∞. 2.22
Let us take v ∈ q with compact support, that is, there exist a, b ∈ , a < b such that
q because if v ∈ q and v k ∈ q
0 is such that v k j 0 if |j| ≥ k 1, v k j vj if |j| ≤ k, then
k q → 0 as k → ∞ Taking v ∈ q
0 in2.22, due to the finite sums and the continuity
of functions f k, ·, we obtain, passing to a limit, that
k∈
ϕ p Δwk − 1Δvk − 1
k∈
V kϕ q wkvk
− λ
k∈
f k, wkvk 0, ∀v ∈ l q
0.
2.23
From the density of l q0 in q, we deduce that the previous equality is fulfilled for all
v ∈ q and, in consequence, w is a critical point of the functional J, that is, w is a solution of
1.10
It remains to show that w / 0
Assuming, on the contrary, that w 0, we conclude that
|u m|∞ |w m|∞ max{|w m k| : k ∈ } −→ 0, as m −→ ∞. 2.24
By F3, for a given ε > 0, there exists δ > 0, such that if |x| < δ then, for every
k
|Fk, x| ≤ ε|x| q ,
Trang 9By 2.24 k such that for
all m > M k it follows that|w m k| < δ Since the maximum value of |w m| is attained at
Then, by2.25, for m > M and every k ∈ :
|Fk, w m k| ≤ ε|w m k| q ,
which implies that
0≤ qJw m q
p
k∈
|Δw m k − 1| p
k∈
V k|w m k| q − λ
k∈
qF k, w m k
k∈
|Δw m k − 1| p
k∈
V k|w m k| q − λ
k∈
f k, w m kw m k
− λ
k∈
qF k, w m k − fk, w m kw m k
≤ Jw m , w m
λqε |w m|q
q ε|w m|q
q
≤Jw m
∗ m q λε q 1
V0 m
q
q
2.28
Since m q , Jw m → 0 and ε is arbitrary, by 2.28 we obtain a
contradiction with Jw m Ju m → c > 0 The proof of the first part is complete.
Now, let u be a nonzero homoclinic solution of problem1.10 Assume that it attains
positive local maximums and/or negative local minimums at infinitely many points k n In particular we can assume that{|k n|} → ∞ In consequence Δ2
p u k n −1uk n ≤ 0 and uk n → 0
From this, multiplying in1.10 by uk n /|uk n|q, we have
λ f k n , u k n uk n
|uk n|q ≥ Δ
2
p u k n − 1uk n
|uk n|q λ f k n , u k n uk n
|uk n|q V k n ≥ V0 > 0. 2.29
By means of conditionF3 we arrive at the following contradiction:
0 λ lim
n→ ∞
f k n , u k n uk n
|uk n|q ≥ V0> 0. 2.30
Suppose now that function u vanishes at infinitely many points l n From condition
F3 we conclude that Δ2
p u l n − 1 0 and, in consequence, ul n − 1ul n 1 < 0 Therefore
Trang 10it has an unbounded sequence of positive local maximums and negative local minimums, in contradiction with the previous assertion
As a direct consequence of the two previous properties, we deduce that, for|k| large enough, function u has constant sign and it is strictly monotone.
To illustrate the optimality of the obtained results, we present in the sequel an example
in which it is pointed out that conditionF2 cannot be removed to deduce the existence result proved inTheorem 1.1
Example 2.5 Let W k > 0 be a T-periodic sequence, W1
p ≥ q > 1 and r > q be fixed Consider problem 1.10 with
f k, t
⎧
⎨
⎩
W kϕ r t if |t| ≤ 1,
It is obvious that condition F1 holds Since r > q we have that condition F3 is trivially fulfilled Concerning to conditionF2, we have that
F k, t
⎧
⎪
⎪
W k |t| r
W k
|t| q
q q − r
It is clear that F
only if 0 < μ ≤ r.
When|t| ≥ 1, the inequality μFk, t ≤ tfk, t holds if and only if either μ q or μ > q
and
|t| q≤ μ
r − q
r
As consequence, the inequality μF k, t ≤ tfk, t for all t / 0 is satisfied if and only if
μ q, that is, condition F2 does not hold
Let us see that this problem has only the trivial solution for small values of the
parameter λ.
Since r > q, it is not difficult to verify that, for 0 < λ < q − 1V0/r − 1W1, the function
λf k, t − V kϕ q t is strictly decreasing for every integer k So, for λ in that situation, we
have that
λf k, t − V kϕ q tt < 0 for all t / 0 and all k ∈ 2.34
Suppose that there is a nontrivial solution u of the considered problem, and moreover
it takes some positive values Let k0be such that uk0 max{uk; k ∈ } > 0 In such a case
we deduce the following contradiction:
0 Δ2
p u k0− 1 − V k0ϕ q uk0 λfk0, u k0 < Δ2
p u k0− 1 ≤ 0. 2.35
Trang 11Analogously it can be verified that the solution u has no negative values on
3 Remarks and Examples
In this section we will consider some examples and remarks on applications and extensions
ofTheorem 1.1to the existence of homoclinic solutions of difference equations of following types:
A Second-order discrete p-Laplacian equations of the form
Δ2
p u k − 1 − V kuk|uk| q−2 λbkuk|uk| r−2 0, 3.1
with r > p ≥ q > 1.
B Higher even-order difference equations A model equation is the fourth-order extended Fisher-Kolmogorov equation
Δ4u k − 2 − aΔ2u k − 1 V kuk|uk| q−2− λbkuk|uk| r−2 0, 3.2
with r > q > 1.
C Second-order difference equations with cubic and quintic nonlinearities of the forms
Δ2u k − 1 − V kuk λb ku3k cku5k 0, 3.3
Δ2
p u k − 1 − akuk λb ku2k cku3k 0, 3.4 arising in mathematical physics and biology
(A) Second-Order Discrete p -Laplacian Equations.
The spectrum of the Dirichlet problem D N for 3.1, subject to Dirichlet boundary conditions
is studied in 17
function, then there exist two positive constants λ0N and λ1N with λ0N ≤ λ1N such that no λ ∈ 0, λ0N is an eigenvalue of problem D N while any λ ∈ λ1N, ∞ is an
eigenvalue of problemD N Moreover, we have
λ1 N ≤ r
N 12|b|∞ ≤ λ0N ≤ λ1N ≤ B
r, q, b, N
... will consider some examples and remarks on applications and extensionsofTheorem 1.1to the existence of homoclinic solutions of difference equations of following types:
A Second-order...
Trang 7Lemma 2.4 Suppose that the assumptions of< /b> Lemma 2.3 hold Then, there exists c > and a ... class="page_container" data-page="10">
it has an unbounded sequence of positive local maximums and negative local minimums, in contradiction with the previous assertion
As a direct consequence of the two