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=Iα ∗ [Q(x)|u|N+α N]Q(x)|u|α N− 1u,x∈ RN. $$\begin{array}{} \displaystyle -{\it\Delta} u+V(x)u=\left(I_{\alpha}\ast [Q(x)|u|^{\frac{N+\alpha}{N}}]\right)Q(x)|u|^{\frac{\alpha}{N}-1}u, \quad x\in \mathbb R^N. \end{array}$$ 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].

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具有加权势和 Hardy-Littlewood-Sobolev 下临界指数的 Choquard 型方程的基态

我们关注一类具有加权势和 Hardy-Littlewood-Sobolev 下临界指数的 Choquard 类型方程 − Δ u+V(x)u=Iα ∗ [Q(x)|u|N+α N]Q(x )|u|α N− 1u,x∈ RN。$$\begin{array}{} \displaystyle -{\it\Delta} u+V(x)u=\left(I_{\alpha}\ast [Q(x)|u|^{\frac{N+ \alpha}{N}}]\right)Q(x)|u|^{\frac{\alpha}{N}-1}u, \quad x\in \mathbb R^N。\end{array}$$ 通过使用变分方法，我们研究了依赖于无穷远处加权势的渐近行为的基态的存在。此外，还考虑了非平凡解的不存在性。特别是，我们对 [D.~Cassani, J. Van Schaftingen 和 JJ Zhang, Groundstates for Choquard type equations with Hardy-Littlewood-Sobolev 下临界指数, Proceedings of the Royal Society of Edinburgh 中提出的一些开放性问题给出了部分答案,