# Existence and nonexistence of solutions to Choquard equations

## TL;DR

This paper investigates the existence and nonexistence of solutions to Choquard equations involving Riesz potentials, establishing conditions for solutions and deriving a Pohožaev identity to identify parameter regimes where solutions cannot exist.

## Contribution

It provides new existence results for ground state solutions of Choquard equations and introduces a Pohožaev identity that delineates nonexistence regions based on parameter conditions.

## Key findings

- Existence of ground state solutions under specific parameter conditions.
- Derivation of a Pohožaev identity for Choquard equations.
- Nonexistence of solutions outside certain parameter ranges.

## Abstract

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