Abstract
In this study, we consider the following quasilinear Choquard equation with singularity
−
Δ
u
+
V
(
x
)
u
−
u
Δ
u
2
+
λ
(
I
α
∗
∣
u
∣
p
)
∣
u
∣
p
−
2
u
=
K
(
x
)
u
−
γ
,
x
∈
R
N
,
u
>
0
,
x
∈
R
N
,
\left\{\begin{array}{ll}-\Delta u+V\left(x)u-u\Delta {u}^{2}+\lambda \left({I}_{\alpha }\ast | u{| }^{p})| u{| }^{p-2}u=K\left(x){u}^{-\gamma },\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\\ u\gt 0,\hspace{1.0em}& x\in {{\mathbb{R}}}^{N},\end{array}\right.
where
I
α
{I}_{\alpha }
is a Riesz potential,
0
<
α
<
N
0\lt \alpha \lt N
, and
N
+
α
N
<
p
<
N
+
α
N
−
2
\displaystyle \frac{N+\alpha }{N}\lt p\lt \displaystyle \frac{N+\alpha }{N-2}
, with
λ
>
0
\lambda \gt 0
. Under suitable assumption on
V
V
and
K
K
, we research the existence of positive solutions of the equations. Furthermore, we obtain the asymptotic behavior of solutions as
λ
→
0
\lambda \to 0
.
Multiplicity and concentration behaviour of solutions for a fractional Choquard equation with critical growth