# Ground state solutions for a nonlinear Choquard equation

## TL;DR

This paper establishes the existence of ground state solutions for a nonlinear Choquard equation using variational methods, addressing different spatial dimensions and specific nonlinearities, contributing to the mathematical understanding of such equations.

## Contribution

It proves the existence of solutions under broad conditions and explores the case of homogeneous nonlinearities, extending previous results with a variational mountain pass approach.

## Key findings

- Existence of ground state solutions for the Choquard equation.
- Different treatment for cases N=2 and N≥3.
- Solutions obtained via variational mountain pass method.

## Abstract

We discuss the existence of ground state solutions for the Choquard equation $$-\Delta u=(I_\alpha*F(u))F'(u)\quad\quad\quad\text{in }\mathbb R^N.$$ We prove the existence of solutions under general hypotheses, investigating in particular the case of a homogeneous nonlinearity $F(u)=\frac{|u|^p}p$. The cases $N=2$ and $N\ge3$ are treated differently in some steps. The solutions are found through a variational mountain pass strategy. The result presented are contained in the papers with arXiv ID 1212.2027 and 1604.03294