Abstract
AbstractIn this paper, we study the second-order Hamiltonian systems $$ \ddot{u}-L(t)u+\nabla W(t,u)=0, $$
u
¨
−
L
(
t
)
u
+
∇
W
(
t
,
u
)
=
0
,
where $t\in \mathbb{R}$
t
∈
R
, $u\in \mathbb{R}^{N}$
u
∈
R
N
, L and W depend periodically on t, 0 lies in a spectral gap of the operator $-d^{2}/dt^{2}+L(t)$
−
d
2
/
d
t
2
+
L
(
t
)
and $W(t,x)$
W
(
t
,
x
)
is locally superquadratic. Replacing the common superquadratic condition that $\lim_{|x|\rightarrow \infty }\frac{W(t,x)}{|x|^{2}}=+\infty $
lim
|
x
|
→
∞
W
(
t
,
x
)
|
x
|
2
=
+
∞
uniformly in $t\in \mathbb{R}$
t
∈
R
by the local condition that $\lim_{|x|\rightarrow \infty }\frac{W(t,x)}{|x|^{2}}=+\infty $
lim
|
x
|
→
∞
W
(
t
,
x
)
|
x
|
2
=
+
∞
a.e. $t\in J$
t
∈
J
for some open interval $J\subset \mathbb{R}$
J
⊂
R
, we prove the existence of one nontrivial homoclinic soluiton for the above problem.
Funder
National Natural Science Foundation of China
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory,Analysis