Existence and nonexistence of solutions to Choquard equations

Abstract

In this paper, we establish the existence of ground state solutions for Choquard equations \begin{equation}\label{eq 1} - Δu + u = q\,(I_α\ast |u|^p) |u|^{q - 2} u+p\,(I_α\ast |u|^q) |u|^{p - 2} u\quad {\rm in }\quad \mathbb{R}^N, \end{equation} where $N \ge 3$, $α\in (0, N)$, $I_α: \mathbb{R}^N \to \mathbb{R}$ is the Riesz potential, $p,\,q >0$ satisfying that \begin{equation}\label{eq 2} \frac{2(N+α)}{N}<p+q< \frac{2(N+α)}{N-2}. \end{equation} Moreover, we prove a Pohožaev type identity for this Choquard equation, which implies the non-existence result for the problem when $(p,q)$ does not satisfy the above condition.