Groundstates for Choquard type equations with weighted potentials and Hardy–Littlewood–Sobolev lower critical exponent

Advances in Nonlinear Analysis

article

Shuai Zhou¹  Zhisu Liu²  Jianjun Zhang¹ 

[1] College of Mathematics and Statistics, Chongqing Jiaotong University;Center for Mathematical Sciences, China University of Geosciences

关键词: Ground states;    Choquard equation;    Hardy–Littlewood–Sobolev inequality;    critical growth;   

DOI  :  10.1515/anona-2020-0186

学科分类：社会科学、人文和艺术（综合）

来源: De Gruyter

【 摘 要 】

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].

【 授权许可】

CC BY