CUBO A Mathematical Journal
Vol.19, No¯ 03, (43–55). October 2017

Weak homoclinic solutions to discrete nonlinear problems of
Kirchhoff type with variable exponents

Aboudramane Guiro, Idrissa Ibrango and Stanislas Ouaro

LAboratoire de Mathématiques et Informatique (LAMI)

1 UFR, Sciences et Techniques, Universit Nazi BONI

01 BP 1091 Bobo-Dioulasso, 01 Bobo Dioulasso, Burkina Faso

2 UFR. Sciences Exactes et Appliques, Universit Ouaga

I Pr Joseph KI-ZERBO, 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso

abouguiro@yahoo.fr, ibrango2006@yahoo.fr, ibrango2006@yahoo.fr, ouaro@yahoo.fr

ABSTRACT

In this paper, we prove the existence of weak homoclinic solutions for discrete nonlinear

problems of Kirchhoff type. The proof of the main result is based on a minimization

method. As extension, we prove the existence result of weak homoclinic solutions for

more general data depending on the solutions.

RESUMEN

En este art́ıculo, probamos la existencia de soluciones homocĺınicas débiles para prob-

lemas discretos no-lineales de tipo Kirchhoff. La demostración del resultado principal

está basado en un método de minimización. Como extensión, probamos la existencia de

soluciones homocĺınicas débiles para datos más generales dependiendo de las soluciones.

2010 AMS Mathematics Subject Classification: 47A75, 35B38, 35P30, 34L05, 34L30.



44 Aboudramane Guiro, Idrissa Ibrango and Stanislas Ouaro CUBO
19, 3 (2017)

1 Introduction

In this paper, we study the following nonlinear discrete anisotropic problem

(P)






−M(A(k − 1, ∆u(k − 1)))∆(a(k − 1, ∆u(k − 1))) + |u(k)|p(k)−2u(k) = f(k), k ∈ Z

lim
|k|→∞

u(k) = 0,

(1.1)

where ∆u(k) = u(k + 1) − u(k) is the forward difference operator.

Note that difference equations can be seen as a discrete counterpart of partial difference equa-

tions and are usually studied in connection with numerical analysis. In this way, the operator in

problem (1.1)

∆(a(k − 1, ∆u(k − 1)))

can be seen as a discrete counterpart of the anisotropic operator

N∑

i=1

∂

∂xi
a

(
x,

∂

∂xi
u

)
.

The first study in that direction for constant exponents was done by Cabada et al. [2] and for

variable exponent by Mihailescu et al. [10] (see also [3]). In [3], the authors studied the following

problem 




−∆
(
a
(
k − 1, ∆u(k − 1)

))
+ |u(k)|p(k)−2u(k) = f(k), k ∈ Z

lim
|k|→+∞

u(k) = 0.
(1.2)

They proved an existence result of weak homoclinic solution of (1.2).

In this paper we consider the same boundary conditions as in [3], but the function

M(A(k − 1, ∆u(k − 1)))

which appear in the left-hand side of problem (1.1) is more general than the one which appear in

[3]. Indeed, if we take M(t) = 1 in the problem (1.1), we obtain the probem studied by Guiro et

als in [3].

To prove an existence result of problem (1.1), we define other new spaces and new associated norms

and we adapt the classical minimization methods used for the study of anisotropic PDEs. The idea

is to transfer the problem of the existence of solutions for (1.1) into the problem of the existence

of a minimizer for some associated energy functional.

The remaining part of this paper is organized as follows: section 2 is devoted to mathematical

preliminaries. The main existence result is stated and proved in section 3. Finally, in section 4,

we discuss some extensions.



CUBO
19, 3 (2017)

Weak homoclinic solutions to discrete nonlinear problems . . . 45

2 Preliminaries

We use the notations

p+ = sup
k∈Z

p(k) and p− = inf
k∈Z

p(k).

For the data f and a, we assume the following:
{
a(., .) : Z × R −→ R and A(., .) : Z × R −→ R with
a(k, ξ) = ∂

∂ξ
A(k, ξ) and A(k, 0) = 0, ∀k ∈ Z.

(2.1)

|ξ|p(k) ≤ a(k, ξ)ξ ≤ p(k)A(k, ξ), ∀ k ∈ Z and ξ ∈ R. (2.2)

There exists a positive constant C1 such that

|a(k, ξ)| ≤ C1(j(k) + |ξ|
p(k)−1), (2.3)

for all k ∈ Z and ξ ∈ R, where j ∈ Lp
′
(.) with 1

p(k)
+ 1

p′(k)
= 1.

Let us take

f ∈ Lp
′
(.). (2.4)

For ξ, η ∈ R with ξ 6= η, and for almost every k ∈ Z,
(
ai(x, ξ) − ai(x, η)

)
(ξ − η) ≥ 0. (2.5)

Moreover, in this paper, we assume that the function

p : Z → (1, +∞) such that 1 < p− < p(.) < p+ < +∞. (2.6)

We also assume that the function M : (0, +∞) → (0, +∞) is continuous, non decreasing and there
exist two positive reals number B1, B2 such that B1 ≤ B2 and α ≥ 1 with

B1t
α−1 ≤ M(t) ≤ B2t

α−1, for t > 0. (2.7)

Let us define the functional spaces,

Lp(.) =

{
u : Z −→ R such that ρp(.)(u) :=

∑

k∈Z

|u(k)|p(k) < ∞
}

and

W1,p(.)α =

{
u : Z −→ R such that ρ1,α,p(.)(u) :=

∑

k∈Z

|u(k)|p(k) +

( ∑

k∈Z

|∆u(k)|p(k)
)α

< ∞
}
.

We introduce in Lp(.) the Luxemburg norm

||u||p(.) := inf

{
λ > 0 such that

∑

k∈Z

∣∣∣∣
u(k)

λ

∣∣∣∣
p(k)

≤ 1

}
,

and we define, on the space W
1,p(.)
α , the norm

||u||1,α,p(.) := inf

{
λ > 0 such that

∑

k∈Z

∣∣∣∣
u(k)

λ

∣∣∣∣
p(k)

+

( ∑

k∈Z

∣∣∣∣
∆u(k)

λ

∣∣∣∣
p(k))α

≤ 1

}
.



46 Aboudramane Guiro, Idrissa Ibrango and Stanislas Ouaro CUBO
19, 3 (2017)

Example 2.1.

As example of functions which satsfies above assumptions, we have the following.

• A(k, ξ) =
1

p(k)
|ξ|p(k), where a(k, ξ) = |ξ|p(k)−2ξ, ∀k ∈ Z and ξ ∈ R,

• A(k, ξ) =
1

p(k)

((
1+ |ξ|2

)p(k)/2
−1

)
, where a(k, ξ) =

(
1+ |ξ|2

)(p(k)−2)/2
ξ, ∀ k ∈ Z, ξ ∈ R

and

• B(t) = ctα−1, for t > 0, where α ≥ 1 and c > 0.

Remark 2.2. If u ∈ Lp(.), then lim
|k|→+∞

u(k) = 0.

Indeed, if u ∈ Lp(.), then
∑

k∈Z

|u(k)|p(k) < +∞.

Let

S1 = {k ∈ Z; |u(k)| < 1} and S2 = {k ∈ Z; |u(k)| ≥ 1}

S2 is necessary a finite set and |u(k)| < +∞ for any k ∈ S2 since u ∈ Lp(.).
As S2 is a finite set, then

∑

k∈S2

|u(k)|p
+

< +∞.

Therefore ∑

k∈Z

|u(k)|p
+

=
∑

k∈S1

|u(k)|p
+

+
∑

k∈S2

|u(k)|p
+

< +∞.

Thus,

lim
|k|→+∞

u(k) = 0. �

In the sequel, we will use the following result:

Proposition 2.3. ([3], Proposition 2.3). If u ∈ Lp(.) and p+ < ∞, then the following properties
hold:

(1) ||u||p(.) < 1 =⇒ ||u||
p+
p(.)

≤ ρp(.)(u) ≤ ||u||
p−

p(.)
;

(2) ||u||p(.) > 1 =⇒ ||u||
p−
p(.)

≤ ρp(.)(u) ≤ ||u||
p+

p(.)
;

(3) ||u||p(.) < 1 (= 1; > 1) ⇐⇒ ρp(.)(u) < 1 (= 1; > 1).

Theorem 2.4. ([3], Theorem 2.1). Let u ∈ Lp(.) and v ∈ Lq(.) such that 1
p(k)

+ 1
q(k)

= 1 for any

k in Z. Then
∑

k∈Z

|uv| ≤

(
1

p−
+

1

q−

)
||u||p(.)||v||q(.). (2.8)



CUBO
19, 3 (2017)

Weak homoclinic solutions to discrete nonlinear problems . . . 47

Proposition 2.5. ([10], Proposition 2.1). Let q : Z −→ R such that

1 < p− ≤ p+ < q− ≤ q+. (2.9)

Then

Lp(.) ⊂ Lq(.). (2.10)

As in [3], we have the following result.

Proposition 2.6.

(1) ρ1,α,p(.)(u + v) ≤ 2
αp+−1(ρ1,α,p(.)(u) + ρ1,α,p(.)(v)), ∀ u, v ∈ W

1,p(.)
α .

(2) Let u in W
1,p(.)
α .

i) If λ > 1, we have

λp
−

ρ1,α,p(.)(u) ≤ ρ1,α,p(.)(λu) ≤ λ
αp+ρ1,α,p(.)(u). (2.11)

ii) If 0 < λ < 1, we have

λαp
+

ρ1,α,p(.)(u) ≤ ρ1,α,p(.)(λu) ≤ λ
p−ρ1,α,p(.)(u). (2.12)

Theorem 2.7. Let u ∈ W
1,p(.)
α \ {0}. Then ||u||1,α,p(.) = a if and only if ρ1,α,p(.)(u/a) = 1.

Proposition 2.8. If u ∈ W
1,p(.)
α and p

+ < ∞, then the following properties hold:

(1) ||u||1,α,p(.) < 1 =⇒ ||u||
αp+
1,α,p(.)

≤ ρ1,α,p(.)(u) ≤ ||u||
p−

1,α,p(.)
;

(2) ||u||1,α,p(.) > 1 =⇒ ||u||
p−
1,α,p(.)

≤ ρ1,α,p(.)(u) ≤ ||u||
αp+

1,α,p(.)
;

(3) ||u||1,α,p(.) < 1 (= 1; > 1) ⇐⇒ ρ1,α,p(.)(u) < 1 (= 1; > 1).

3 Existence of weak homoclinic solutions

In this section we investigate the existence of weak homoclinic solutions of problem (1.1).

The energy functional corresponding to problem (1.1) is defined as J : W
1,p(.)
α −→ R such that

J(u) = M̂

( ∑

k∈Z

A
(
k − 1, ∆u(k − 1)

))
+
∑

k∈Z

1

p(k)
|u(k)|p(k) −

∑

k∈Z

f(k)u(k), (3.1)

where M̂(t) =

∫t

0

M(s) ds.



48 Aboudramane Guiro, Idrissa Ibrango and Stanislas Ouaro CUBO
19, 3 (2017)

Definition 3.1. A weak homoclinic solution of problem (1.1) is a function u ∈ W
1,p(.)
α such that






M

( ∑

k∈Z

A
(
k − 1, ∆u(k − 1)

)) ∑

k∈Z

a(k − 1, ∆u(k − 1))∆v(k − 1)

+
∑

k∈Z

|u(k)|p(k)−2u(k)v(k) =
∑

k∈Z

f(k)v(k),

(3.2)

for any v ∈ W
1,p(.)
α .

Hence the critical points of functionnal J are the weak solutions for problem (1.1).

The main result is the following:

Theorem 3.2. Assume that hypotheses (2.1)-(2.7) hold. Then, there exists at least one weak

homoclinic solution of problem (1.1).

Proof. We first present some basic properties of the functional J.

Proposition 3.3. The functional J is well defined on W
1,p(.)
α and is of class C

1(W
1,p(.)
α , R) with

the derivative given by






〈J′(u), v〉 = M

( ∑

k∈Z

A
(
k − 1, ∆u(k − 1)

)) ∑

k∈Z

a(k − 1, ∆u(k − 1))∆v(k − 1)

+
∑

k∈Z

|u(k)|p(k)−2u(k)v(k) −
∑

k∈Z

f(k)v(k),

(3.3)

for all u, v ∈ W
1,p(.)
α .

Indeed, we denote by

I(u) = M̂

( ∑

k∈Z

A
(
k − 1, ∆u(k − 1)

))
, L(u) =

∑

k∈Z

1

p(k)
|u(k)|p(k) and Λ(u) =

∑

k∈Z

f(k)u(k).

We have, by using (2.7), that

|I(u)| =

∣∣∣∣
∫
∑

k∈Z

A
(
k − 1, ∆u(k − 1)

)

0

M(t)dt

∣∣∣∣

≤ B2

∣∣∣∣
∫
∑

k∈Z

A
(
k − 1, ∆u(k − 1)

)

0

tα−1dt

∣∣∣∣

≤
B2

α

( ∑

k∈Z

∣∣A(k − 1, ∆u(k − 1))
∣∣
)α

.



CUBO
19, 3 (2017)

Weak homoclinic solutions to discrete nonlinear problems . . . 49

According to (2.1), (2.3) and the discrete Hlder type inequality, we write

∑

k∈Z

∣∣A(k − 1, ∆u(k − 1))
∣∣ ≤

∑

k∈Z

∫∆u(k−1)

0

|a(k − 1, t)|dt

≤ C1
∑

k∈Z

(
j(k − 1) +

1

p(k − 1)
|∆u(k − 1)|p(k−1)−1

)
∆u(k − 1)

≤ C1
∑

k∈Z

j(k − 1)|∆u(k − 1)| +
C1

p−

∑

k∈Z

|∆u(k − 1)|p(k−1)

≤ C1

(
1

q−
+

1

p−

)
||j||q(.)||∆u||p(.) +

C1

p−
||∆u||p(.)

< +∞,

and we deduce that |I(u)| < +∞.
We have

|L(u)| =

∣∣∣∣
∑

k∈Z

1

p(k)
|u(k)|p(k)

∣∣∣∣ ≤
1

p−

∣∣∣∣
∑

k∈Z

|u(k)|p(k)
∣∣∣∣ < +∞,

and

|Λ(u)| =

∣∣∣∣
∑

k∈Z

f(k)u(k)

∣∣∣∣ ≤
∑

k∈Z

|f(k)||u(k)| < +∞.

Clearly, the functionals I, L and Λ are in C1(W
1,p(.)
α , R).

Therefore, the functional J is well defined on W
1,p(.)
α and is of class C

1(W
1,p(.)
α , R).

In what follows we prove (3.3). Let u, v ∈ W
1,p(.)
α . We have






lim
h→0+

I(u + hv) − I(u)

h

= lim
h→0+

M̂

( ∑

k∈Z

A
(
k − 1, ∆u(k − 1) + h∆v(k − 1)

))
− M̂

( ∑

k∈Z

A
(
k − 1, ∆u(k − 1)

))

h

= M

( ∑

k∈Z

A
(
k − 1, ∆u(k − 1)

)) ∑

k∈Z

a
(
k − 1, ∆u(k − 1)

)
∆v(k − 1).

Let us denote gh =
|u(k) + hv(k)|p(k) − |u(k)|p(k)

p(k)h
.

We have
∑

k∈Z

|gh| ≤
1

p−h

∑

k∈Z

|u(k)|p(k) +
1

p−h

∑

k∈Z

|u(k) + hv(k)|p(k) < +∞.



50 Aboudramane Guiro, Idrissa Ibrango and Stanislas Ouaro CUBO
19, 3 (2017)

Thus

lim
h→0+

L(u + hv) − L(u)

h
= lim

h→0+

∑

k∈Z

|u(k) + hv(k)|p(k) − |u(k)|p(k)

p(k)h

=
∑

k∈Z

lim
h→0+

|u(k) + hv(k)|p(k) − |u(k)|p(k)

p(k)h

=
∑

k∈Z

|u(k)|p(k)−2u(k)v(k).

And

lim
h→0+

Λ(u + hv) − Λ(u)

h
= lim

h→0+

∑

k∈Z

f(k)
(
u(k) + hv(k)

)
− f(k)u(k)

h

=
∑

k∈Z

f(k)v(k). �

Proposition 3.4. The functional J is weakly lower semi-continuous.

Indeed, by (2.1), (2.5) and (2.7) we have that J is convex. Thus, it is enough to show that J is

lower semi-continuous. For this, we fix u ∈ W
1,p(.)
α and ǫ > 0. Since J is convex, we deduce that

for any v ∈ W
1,p(.)
α ,

J(v) ≥ J(u) + 〈J′(u), v − u〉

≥ J(u) + R(u, v) + S(u, v) + T(u, v),

with

R(u, v) = M

( ∑

k∈Z

A
(
k − 1, ∆u(k − 1)

)) ∑

k∈Z

a(k − 1, ∆u(k − 1))
(
∆v(k − 1) − ∆u(k − 1)

)

and

S(u, v) =
∑

k∈Z

|u(k)|p(k)−2u(k)
(
v(k) − u(k)

)
and T(u, v) =

∑

k∈Z

f(k)
(
u(k) − v(k)

)
.

Using the Hlder type inequality there exist tree non negative constants C3, C4 and C5 such that

R(u, v) ≥ −M

( ∑

k∈Z

A
(
k − 1, ∆u(k − 1)

)) ∑

k∈Z

|a(k − 1, ∆u(k − 1))||∆v(k − 1) − ∆u(k − 1)|

≥ −C3||∆u − ∆v||p(.)

≥ −C3||u − v||1,α,p(.) (3.4)

and

T(u, v) ≥ −C4||u − v||1,α,p(.). (3.5)



CUBO
19, 3 (2017)

Weak homoclinic solutions to discrete nonlinear problems . . . 51

Also

S(u, v) ≥ −
∑

k∈Z

|u(k)|p(k)−1|v(k) − u(k)|

≥ −
( 1
p−

+
1

(p′)−

)∣∣∣∣|u|p(.)−1
∣∣∣∣
p′(.)

||u − v||p(.)

≥ −C5||u − v||1,α,p(.). (3.6)

Then, combining (3.4), (3.5) and (3.6), we get

J(v) ≥ J(u) − C6||u − v||1,α,p(.) (3.7)

with C6 = C3 + C4 + C5.

Finally for all v ∈ W
1,p(.)
α with ||v − u||1,α,p(.) < δ =

ǫ

C6
, we get

J(v) ≥ J(u) − ǫ.

Then J is lower semi-continuous and by corollary III.8 in [1], we conclude that J is weakly lower

semi-continuous. �

Proposition 3.5. The functional J is coercive and bounded from below.

Indeed, we have

J(u) = M̂

( ∑

k∈Z

A
(
k − 1, ∆u(k − 1)

))
+
∑

k∈Z

1

p(k)
|u(k)|p(k) −

∑

k∈Z

f(k)u(k)

≥
B1

α

( ∑

k∈Z

A
(
k − 1, ∆u(k − 1)

))α
+

1

p+

∑

k∈Z

|u(k)|p(k) − C7||f||p′(.)||u||p(.)

≥
B1

α

( ∑

k∈Z

1

p(k − 1)

∣∣∆u(k − 1)
∣∣p(k−1)

)α
+

1

p+

∑

k∈Z

|u(k)|p(k) − C′7||u||1,α,p(.)

≥
B1

α(p+)α

( ∑

k∈Z

∣∣∆u(k)
∣∣p(k)

)α
+

1

p+

∑

k∈Z

|u(k)|p(k) − C′7||u||1,α,p(.)

≥ min

(
B1

α(p+)α
;
1

p+

)(( ∑

k∈Z

∣∣∆u(k)
∣∣p(k)

)α
+
∑

k∈Z

|u(k)|p(k)
)
− C′7||u||1,α,p(.)

≥ min

(
B1

α(p+)α
;
1

p+

)
ρ1,α,p(.)(u) − C

′

7||u||1,α,p(.).

To prove the coerciveness of the functional J, we may assume that ||u||1,α,p(.) > 1 and, using

Proposition 2.8, we deduce from the above inequality that

J(u) ≥ min

(
B1

α(p+)α
;
1

p+

)
||u||

p
−

1,α,p(.)
− C′7||u||1,α,p(.).

Thus,

J(u) −→ +∞ as ||u||1,α,p(.) −→ +∞,



52 Aboudramane Guiro, Idrissa Ibrango and Stanislas Ouaro CUBO
19, 3 (2017)

namely J is coercive.

Besides, for ||u||1,α,p(.) ≤ 1, we have

J(u) ≥ min

(
B1

α(p+)α
;
1

p+

)
ρ1,α,p(.)(u) − C

′

7||u||1,α,p(.)

≥ −C′7||u||1,α,p(.)

> −∞.

Thus J is bounded below. �

Since J is proper, weakly lower semi-continuous and coercive on W
1,p(.)
α , using the relation between

critical points of J and problem (1.1), we deduce that J has a minimizer which is a weak homoclinic

solution of (1.1). �

4 An extension

In this section we consider the following problem

(Pu)






−M(A(k − 1, ∆u(k − 1)))∆(a(k − 1, ∆u(k − 1)))

+|u(k)|p(k)−2u(k) = f(k, u(k)), k ∈ Z

lim
|k|→∞

u(k) = 0,

(4.1)

where we assume that






f(k, .) : R −→ R is a continuous function for all k in Z;
c : Z −→ [0; +∞) is such that c+ < +∞ and |f(k, t)| ≤ c(k)|t|β(k)−1,
with β : Z −→ R such that 1 < p− ≤ p+ < β− ≤ β+.

(4.2)

For any u ∈ W
1,p(.)
α , the energy functional corresponding to (4.1) is

J(u) = M̂

( ∑

k∈Z

A
(
k − 1, ∆u(k − 1)

))
+
∑

k∈Z

1

p(k)
|u(k)|p(k) −

∑

k∈Z

F
(
k, u(k)

)
, (4.3)

where F
(
k, u(k)

)
=

∫u(k)

0

f(k, t)dt.



CUBO
19, 3 (2017)

Weak homoclinic solutions to discrete nonlinear problems . . . 53

Definition 4.1. A weak homoclinic solution of problem (4.1) is a function u ∈ W
1,p(.)
α such that






M

( ∑

k∈Z

A
(
k − 1, ∆u(k − 1)

)) ∑

k∈Z

a(k − 1, ∆u(k − 1))∆v(k − 1)

+
∑

k∈Z

|u(k)|p(k)−2u(k)v(k) =
∑

k∈Z

f
(
k, u(k)

)
v(k),

(4.4)

for any v ∈ W
1,p(.)
α .

We have the following result:

Theorem 4.2. Under assumptions (2.1)-(2.7) and (4.2), the problem (4.1) has at least one weak

solution.

Proof. Let h(u) =
∑

k∈Z

F
(
k, u(k)

)
, therefore h′ : W

1,p(.)
α −→ W

1,p(.)
α is completly continuous

and thus, h is weakly lower semi-continuous. Then J ∈ C1
(
W

1,p(.)
α ; R

)
and is weakly lower semi-

continuous.

On the other hand, for all u, v ∈ W
1,p(.)
α , we have

lim
δ→0+

h(u + δv) − h(u)

δ
= lim

δ→0+

∑

k∈Z

F
(
k, u(k) + δv(k)

)
− F

(
k, u(k)

δ
,

and since

∑

k∈Z

∣∣∣∣
F
(
k, u(k) + δv(k)

)
− F

(
k, u(k)

)

δ

∣∣∣∣ ≤
c+

δ

∑

k∈Z

( ∫u(k)+δv(k)

0

|t|β(k)−1dt +

∫u(k)

0

|t|β(k)−1dt

)

≤
c+

δβ−

∑

k∈Z

(
|u(k) + δv(k)|β(k) + |u(k)|β(k)

)

< +∞,

we obtain

lim
δ→0+

h(u + δv) − h(u)

δ
=

∑

k∈Z

lim
δ→0+

F
(
k, u(k) + δv(k)

)
− F

(
k, u(k)

δ

=
∑

k∈Z

f
(
k, u(k)

)
v(k).

Consequently,





〈J′(u), v〉 = M

( ∑

k∈Z

A
(
k − 1, ∆u(k − 1)

)) ∑

k∈Z

a(k − 1, ∆u(k − 1))∆v(k − 1)

+
∑

k∈Z

|u(k)|p(k)−2u(k)v(k) −
∑

k∈Z

f
(
k, u(k)

)
v(k),

(4.5)



54 Aboudramane Guiro, Idrissa Ibrango and Stanislas Ouaro CUBO
19, 3 (2017)

for all u, v ∈ W
1,p(.)
α .

A critical point to J, i.e. a point u ∈ W
1,p(.)
α such that

〈J′(u), v〉 = 0 for all v ∈ W1,p(.)α

is a weak solution to (4.1). To end the proof of Theorem 4.2, we have to prove that J is coercive

and bounded from below.

For u ∈ W
1,p(.)
α such that ||u||1,α,p(.) > 1, we have

J(u) ≥ min

(
B1

α(p+)α
;
1

p+

)
||u||

p−

1,α,p(.)
−
∑

k∈Z

F
(
k, u(k)

)

≥ min

(
B1

α(p+)α
;
1

p+

)
||u||

p−

1,α,p(.)
−
∑

k∈Z

∫u(k)

0

|f(k, t)|dt

≥ min

(
B1

α(p+)α
;
1

p+

)
||u||

p−

1,α,p(.)
−

c+

β−

∑

k∈Z

|u(k)|β(k)

≥ min

(
B1

α(p+)α
;
1

p+

)
||u||

p−

1,α,p(.)
−

c+K

β−

where K is a positive constant. As p− > 1, then J is coercive.

On the other hand, for u ∈ W
1,p(.)
α such that ||u||1,α,p(.) < 1, we have

J(u) ≥ min

(
B1

α(p+)α
;
1

p+

)
||u||

αp+

1,α,p(.)
−

c+K

β−

≥ −
c+K

β−

> −∞.

Therefore, J is bounded from below. �

References

[1] H. Brezis; Analyse Fonctionnelle: Theorie et Applications. Paris, Masson, 1983.

[2] A. Cabada, C. Li, S. Tersian; On homoclinic solutions of a semilinear p-Laplacian difference

equation with periodic coefficients. Adv. Difference Equ. 2010, Art. ID 195376, 17 pp.

[3] A. Guiro, B. Kon and S. Ouaro; Weak homoclinic solutions of anisotropic difference equation

with variable exponents. Adv. Difference Equ. 154 (2013), 13 pp.

[4] A. Guiro, B. Kon and S. Ouaro; Weak heteroclinic solutions of anisotropic difference equation

with variable exponents. Electron. J. Diff. Equ. 225 (2013), 1-9.



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19, 3 (2017)

Weak homoclinic solutions to discrete nonlinear problems . . . 55

[5] B. Kon and S. Ouaro; Weak solutions for anisotropic discrete boundary value problems. J.

Difference Equ. Appl. 17, N.10, (2011), 1537-1547.

[6] B. Kon, S. Ouaro and S. Traor; Weak solutions for anisotropic nonlinear elliptic equations with

variable exponent. Electron. J. Diff. Equ. 144 (2009), 1-11.

[7] M. Mihailescu, P. Pucci and V. Radulescu; Nonhomogeneous boundary value problems in

anisotropic Sobolev spaces. C. R. Acad. Sci. Paris, Ser. I 345 (2007), 561-566.

[8] M. Mihailescu, P. Pucci and V. Radulescu; Eigenvalue problems for anisotropic quasilinear

elliptic equations with variable exponent. J. Math. Anal. Appl. 340 (2008), 687-698.

[9] M. Mihailescu, V. Radulescu and S. Tersian; Eigenvalue problems for anisotropic discrete

boundary value problems. J. Difference Equ. Appl. 15(6) (2009), 557-567.

[10] M. Mihailescu, V. Radulescu and S. Tersian; Homoclinic solutions of difference equations with

variable exponents. Topol. Methods Nonlinear Anal. 38 (2011) 2, 277-289.


	Introduction
	Preliminaries
	Existence of weak homoclinic solutions
	An extension