Abstract
In this paper, we study the following fractional Kirchhoff type equation
{
(
a
+
b
∫
R
N
∫
R
N
|
u
(
x
)
−
u
(
y
)
|
p
|
x
−
y
|
N
+
p
s
d
x
d
y
)
(
−
Δ
)
p
s
u
=
|
u
|
q
−
2
u
ln
|
u
|
2
+
λ
u
γ
,
i
n
Ω
,
u
>
0
,
i
n
Ω
,
u
=
0
,
i
n
R
N
∖
Ω
,
where
Ω
⊂
R
N
is a bounded domain with Lipschitz boundary,
0
<
s
<
1
<
p
,
0
<
γ
<
1
,
a
>
0
,
b
≥
0
,
N
>
p
s
,
2
p
<
q
<
q
+
2
<
p
s
∗
,
p
s
∗
=
N
p
N
−
p
s
is the fractional critical exponent,
λ
>
0
is a real parameter. By using the critical point theory for nonsmooth functionals and analytic techniques, the existence and multiplicity of positive solutions are obtained.