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

• Published in: Proceedings of the Royal Society of Edinburgh. Section A. Mathematics 2020-06, Vol.150 (3), p.1377-1400
• Main Authors: Cassani, Daniele, Van Schaftingen, Jean, Zhang, Jianjun
• Format: Article
• Language: English
• Subjects: Ground state
• ISSN: 0308-2105; 1473-7124
• DOI: 10.1017/prm.2018.135

For the Choquard equation, which is a nonlocal nonlinear Schrödinger type equation, $$-\Delta u+V_{\mu, \nu} u=(I_\alpha\ast \vert u \vert ^{({N+\alpha})/{N}}){ \vert u \vert }^{{\alpha}/{N}-1}u,\quad {\rm in} \ {\open R}^N, $$ where $N\ges 3$ , V μ,ν :ℝ N  → ℝ is an external potential defined for μ, ν > 0 and x  ∈ ℝ N by V μ,ν ( x ) = 1 − μ/(ν 2  + | x | 2 ) and $I_\alpha : {\open R}^N \to 0$ is the Riesz potential for α ∈ (0, N ), we exhibit two thresholds μ ν , μ ν  > 0 such that the equation admits a positive ground state solution if and only if μ ν