Existence and concentration of ground states to fractional Choquard equations

Abstract

In this paper, we study the nonlinear fractional Choquard equation \begin{equation}\label{eq:0.1a} (-Δ)^su+Vu=(|x|^{-γ}*|u|^2)u \quad {\rm in} \quad \mathbb{R}^N, \end{equation} where $0<γ<4$, $0<s<1$, $N\geq 4$ and $V\in C^1(\mathbb{R}^N)$ is a positive potential. Set $s_0=\frac γ{4}$. Under suitable assumptions on $V$, we prove that the equation admits a nonnegative ground state solution for $s\in(s_0,1)$, whereas no ground state solution exists for $0<s\le s_0$. Furthermore, we show that any ground state solution $u_s$ blows up and concentrates at a minimum point of $V$ as $s\downarrow s_0$. Finally, up to a subsequence, the ground state solution $u_s$ converges to a ground state solution of the classical Choquard equation as $s\uparrow 1$.