Author:
Deng Hai-yun,Lin Xiao-yan,He Yu-bo
Abstract
AbstractIn this paper, we consider a $(\phi _{1},\phi _{2})$
(
ϕ
1
,
ϕ
2
)
-Laplacian system as follows: $$\begin{aligned} \textstyle\begin{cases} \Delta \phi _{1} (\Delta u(t-1) )+\nabla _{u} F(t,u(t),v(t))=0, \\ \Delta \phi _{2} (\Delta v(t-1) )+\nabla _{v} F(t,u(t),v(t))=0, \end{cases}\displaystyle \end{aligned}$$
{
Δ
ϕ
1
(
Δ
u
(
t
−
1
)
)
+
∇
u
F
(
t
,
u
(
t
)
,
v
(
t
)
)
=
0
,
Δ
ϕ
2
(
Δ
v
(
t
−
1
)
)
+
∇
v
F
(
t
,
u
(
t
)
,
v
(
t
)
)
=
0
,
where $F(t,u(t),v(t))=-K(t,u(t),v(t))+W(t,u(t),v(t))$
F
(
t
,
u
(
t
)
,
v
(
t
)
)
=
−
K
(
t
,
u
(
t
)
,
v
(
t
)
)
+
W
(
t
,
u
(
t
)
,
v
(
t
)
)
is T-periodic in t. By using the mountain pass theorem, we obtain that the $(\phi _{1},\phi _{2})$
(
ϕ
1
,
ϕ
2
)
-Laplacian system has at least one periodic solution if W is asymptotically $(p,q)$
(
p
,
q
)
-linear at infinity. Our results improve and extend some known works.
Funder
The Natural Science Foundation of Hunan Province, China
Research Foundation of Education Bureau of Hunan Province, China
Publisher
Springer Science and Business Media LLC