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

Abstract

For the Choquard equation, which is a nonlocal nonlinear Schr\"odinger type equation, $ -\Delta u+V_{\mu,\nu} u=(I_\alpha\ast |u|^{\frac{N+\alpha}{N}}){|u|}^{\frac{\alpha}{N}-1}u$, in $\mathbb{R}^N$ where $N\ge 3$, $V_{\mu, \nu} : \mathbb{R}^N \to \mathbb{R}$ is an external potential defined for $\mu,\nu\in\mathbb{R}$ and $x \in \mathbb{R}^N$ by $V_{\mu, \nu} (x)=1-\mu/(\nu^2 + |x|^2)$ and $I_\alpha : \mathbb{R}^N \to 0$ is the Riesz potential for $\alpha\in (0,N)$, we exhibit two thresholds $\mu_{\nu},\mu^{\nu}>0$ such that the equation admits a positive ground state solution if and only if $\mu_{\nu}<\mu<\mu^{\nu}$ and no ground state solution exists for $\mu<\mu_{\nu}$. Moreover, if $\mu>\max\left\{\mu^{\nu},\frac{N^2(N-2)}{4(N+1)}\right\}$, then equation still admits a sign changing ground state solution provided $N\ge4$ or in dimension $N=3$ if in addition $\frac{3}{2}<\alpha<3$ and ker$(-\Delta + V_{\mu, \nu}) = \{0\}$, namely in the non-resonant case.