Abstract
We study the normalized ground state solutions for the following mass critical Kirchhoff equations with inhomogeneous interactions
{
−
(
a
+
b
∫
R
N
|
∇
u
|
2
d
x
)
Δ
u
+
V
(
x
)
u
=
λ
u
+
β
m
(
x
)
u
p
+
1
in
R
N
,
∫
R
N
|
u
|
2
d
x
=
1
,
where
a
≥
0
,
b
>
0
,
β
>
0
,
p
=
8
N
,
0
m
(
x
)
≤
1
and the function
V
(
x
)
is a trapping potential in
R
N
(
N
=
1
,
2
,
3
)
. We prove that there exists
β
∗
>
0
such that the constraint minimizers exist if the interaction strength
β
satisfies
β
≤
β
∗
when
a
>
0
and if
β
β
∗
when
a
=
0
. Further, we investigate the limit behavior of minimizers as
a
→
0
for
β
=
β
∗
. We also prove that all the mass of the minimizers concentrate at a global minimum point
x
0
of
V
(
x
)
as
a
→
0
, where
x
0
is also a global maximum point of
m
(
x
)
. Specially, we find that the limit behavior is related to the behavior of the functions
V
(
x
)
and
m
(
x
)
at their critical points.