Affiliation:
1. School of Mathematics and Computational Science, Hunan University of Science and Technology , Xiangtan , Hunan 411201 , P. R. China
2. School of Mathematics and Computational Science, Huaihua University , Huaihua , Hunan 418008 , P. R. China
3. School of Mathematics, Anhui University of Science and Technology , Huainan , Anhui 232001 , P. R. China
Abstract
Abstract
We study the following fractional Choquard equation
(
−
Δ
)
s
u
+
u
∣
x
∣
θ
=
(
I
α
*
F
(
u
)
)
f
(
u
)
,
x
∈
R
N
,
{\left(-\Delta )}^{s}u+\frac{u}{{| x| }^{\theta }}=({I}_{\alpha }* F\left(u))f\left(u),\hspace{1em}x\in {{\mathbb{R}}}^{N},
where
N
⩾
3
N\geqslant 3
,
s
∈
1
2
,
1
s\in \left(\phantom{\rule[-0.75em]{}{0ex}},\frac{1}{2},1\right)
,
α
∈
(
0
,
N
)
\alpha \in \left(0,N)
,
θ
∈
(
0
,
2
s
)
\theta \in \left(0,2s)
, and
I
α
{I}_{\alpha }
is the Riesz potential. The main purpose of this article is twofold. We first study the regularity of weak solutions for the aforementioned equation with critical nonlinearity, which extends the results of
θ
=
0
\theta =0
in Moroz-Van Schaftingen [Existence of groundstates for a class of nonlinear Choquardequations, Trans. Amer. Math. Soc. 367 (2015), no. 9, 6557–6579]. Then, as an application of the regularity results, we establish the existence of ground state solutions for above equation with the nonlinearity involving embedding top and bottom indices, which is related to the Hardy-Littlewood-Sobolev inequality and singular term
1
∣
x
∣
θ
\frac{1}{{| x| }^{\theta }}
. It is worth noting that our approach is not involving the concentration-compactness principle.