Abstract
In this paper, we consider the following nonlinear Klein–Gordon–Maxwell system with a steep potential well
{
−
Δ
u
+
(
λ
a
(
x
)
+
1
)
u
−
μ
(
2
ω
+
ϕ
)
ϕ
u
=
f
(
x
,
u
)
,
in
R
3
,
Δ
ϕ
=
μ
(
ω
+
ϕ
)
u
2
,
in
R
3
,
where
ω
>
0
is a constant,
μ
and
λ
are positive parameters,
f
∈
C
(
R
3
×
R
,
R
)
and the nonlinearity
f
satisfies the Ambrosetti–Rabinowitz condition. We use parameter-dependent compactness lemma to prove the existence of nontrivial solution for
μ
small and
λ
large enough, then explore the asymptotic behavior as
μ
→
0
and
λ
→
∞
. Moreover, we also use truncation technique to study the existence and asymptotic behavior of positive solution of Klein–Gordon–Maxwell system when
f
(
u
)
:=
|
u
|
q
−
2
u
where
2
<
q
<
4
.
Funder
National Natural Science Foundation of China