Affiliation:
1. Department of Basic Teaching and Research, Yango University, Fuzhou , Fujian 350015 , China
2. College of Mathematics and Informatics & FJKLMAA, Fujian Normal University, Qishan Campus , Fuzhou 350117 , China
Abstract
Abstract
We investigate a fractional Schrödinger-Poisson system with strong singularity as follows:
(
−
Δ
)
s
u
+
V
(
x
)
u
+
λ
ϕ
u
=
f
(
x
)
u
−
γ
,
x
∈
R
3
,
(
−
Δ
)
t
ϕ
=
u
2
,
x
∈
R
3
,
u
>
0
,
x
∈
R
3
,
\left\{\begin{array}{ll}{\left(-\Delta )}^{s}u+V\left(x)u+\lambda \phi u=f\left(x){u}^{-\gamma },& x\in {{\mathbb{R}}}^{3},\\ {\left(-\Delta )}^{t}\phi ={u}^{2},& x\in {{\mathbb{R}}}^{3},\\ u\gt 0,& x\in {{\mathbb{R}}}^{3},\end{array}\right.
where
0
<
s
≤
t
<
1
0\lt s\le t\lt 1
with
4
s
+
2
t
>
3
4s+2t\gt 3
,
λ
>
0
\lambda \gt 0
and
γ
>
1
\gamma \gt 1
. When
V
V
and
f
f
satisfy certain conditions, existence and uniqueness of positive solution
u
λ
{u}_{\lambda }
are established via variational method and Nehari method. We also describe the asymptotic behaviour of
u
λ
{u}_{\lambda }
as
λ
→
0
\lambda \to 0
.