Abstract
We are concerned with a class of Choquard type equations with weighted potentials and Hardy–Littlewood–Sobolev lower critical exponent
−
Δ
u
+
V
(
x
)
u
=
I
α
∗
[
Q
(
x
)
|
u
|
N
+
α
N
]
Q
(
x
)
|
u
|
α
N
−
1
u
,
x
∈
R
N
.
$$\begin{array}{}
\displaystyle
-{\it\Delta} u+V(x)u=\left(I_{\alpha}\ast [Q(x)|u|^{\frac{N+\alpha}{N}}]\right)Q(x)|u|^{\frac{\alpha}{N}-1}u, \quad x\in \mathbb R^N.
\end{array}$$
By using variational approaches, we investigate the existence of groundstates relying on the asymptotic behaviour of weighted potentials at infinity. Moreover, non-existence of non-trivial solutions is also considered. In particular, we give a partial answer to some open questions raised in [D.~Cassani, J. Van Schaftingen and J. J. Zhang, Groundstates for Choquard type equations with Hardy-Littlewood-Sobolev lower critical exponent, Proceedings of the Royal Society of Edinburgh, Section A Mathematics, 150(2020), 1377–1400].