Abstract
We consider the following Kirchhoff equation with critical Sobolev exponent
(K)
{
(
a
+
b
∫
R
N
|
(
−
Δ
)
s
2
u
|
2
d
x
)
(
−
Δ
)
s
u
=
|
u
|
2
s
∗
−
2
u
+
μ
g
(
x
)
,
x
∈
R
N
,
u
∈
D
s
,
2
(
R
N
)
,
where
0
s
≤
1
,
N
>
2
s
,
a
>
0
,
b
∈
R
∖
{
0
}
,
μ
is a non-negative parameter,
g
∈
L
2
N
N
+
2
s
(
R
N
)
is a positive function,
2
s
∗
=
2
N
N
−
2
s
is the critical Sobolev exponent. We show that
(K)
has at least two positive solutions by the variational method. Particularly, in contrast to most existing results in the literature, our method can be applied to deal with wide ranges of
s
and
N
.nd
N
.