AbstractIn this paper, we consider the following fractional Kirchhoff problem with strong singularity: $$ \textstyle\begin{cases} (1+ b\int _{\mathbb{R}^{3}}\int _{\mathbb{R}^{3}} \frac{ \vert u(x)-u(y) \vert ^{2}}{ \vert x-y \vert ^{3+2s}}\,\mathrm{d}x \,\mathrm{d}y )(-\Delta )^{s} u+V(x)u = f(x)u^{-\gamma }, & x \in \mathbb{R}^{3}, \\ u>0,& x\in \mathbb{R}^{3}, \end{cases} $$
{
(
1
+
b
∫
R
3
∫
R
3
|
u
(
x
)
−
u
(
y
)
|
2
|
x
−
y
|
3
+
2
s
d
x
d
y
)
(
−
Δ
)
s
u
+
V
(
x
)
u
=
f
(
x
)
u
−
γ
,
x
∈
R
3
,
u
>
0
,
x
∈
R
3
,
where $(-\Delta )^{s}$
(
−
Δ
)
s
is the fractional Laplacian with $0< s<1$
0
<
s
<
1
, $b>0$
b
>
0
is a constant, and $\gamma >1$
γ
>
1
. Since $\gamma >1$
γ
>
1
, the energy functional is not well defined on the work space, which is quite different with the situation of $0<\gamma <1$
0
<
γ
<
1
and can lead to some new difficulties. Under certain assumptions on V and f, we show the existence and uniqueness of a positive solution $u_{b}$
u
b
by using variational methods and the Nehari manifold method. We also give a convergence property of $u_{b}$
u
b
as $b\rightarrow 0$
b
→
0
, where b is regarded as a positive parameter.
The Nehari manifold for a fractional Laplacian equation involving critical nonlinearities