Affiliation:
1. Hunan University of Science and Technology
Abstract
In this paper, we prove the existence of a positive ground state
solution to the following coupled system involving nonlinear Schrödinger
equations:
{
−
Δ
u
+
V
1
(
x
)
u
=
f
1
(
x
,
u
)
+
λ
(
x
)
v
,
x
∈
R
2
,
−
Δ
v
+
V
2
(
x
)
v
=
f
2
(
x
,
v
)
+
λ
(
x
)
u
,
x
∈
R
2
,
where
λ
,
V
1
,
V
2
∈
C
(
R
2
,
(
0
,
+
∞
)
)
and
V
1
(
x
)
have
critical exponential growth in the sense of Trudinger–Moser inequality. The
potentials
V
1
(
x
)
and
V
2
(
x
)
satisfy a condition involving the
coupling term
λ
(
x
)
, namely
0
<
λ
(
x
)
≤
λ
0
V
1
(
x
)
V
2
(
x
)
. We use non-Nehari
manifold, Lions's concentration compactness and strong maximum principle to
get a positive ground state solution. Moreover, by using a bootstrap
regularity lifting argument and
L
q
-estimates we get regularity and
asymptotic behavior. Our results improve and extend the previous results.