A guide to the Choquard equation

TL;DR

This paper provides a comprehensive survey of the existence, properties, and various results related to solutions of the Choquard equation and its variants, highlighting both historical and recent developments.

Contribution

It offers an extensive overview of theoretical and applied research on the Choquard equation, consolidating known results and recent advances in the field.

Findings

Summarizes existence results for solutions to the Choquard equation.

Discusses properties and extensions of the equation.

Highlights recent developments and open problems.

Abstract

We survey old and recent results dealing with the existence and properties of solutions to the Choquard type equations $$ -\Delta u + V(x)u = \bigl(|x|^{-(N-\alpha)} * |u|^p\bigr)|u|^{p - 2} u \qquad \text{in $\mathbb{R}^N$}, $$ and some of its variants and extensions.