Existence of groundstates for a class of nonlinear Choquard equations in the plane

TL;DR

This paper establishes the existence of groundstate solutions for a class of nonlinear Choquard equations in the plane, expanding understanding of such equations with general nonlinearities and Riesz potentials.

Contribution

It proves the existence of nontrivial groundstate solutions for nonlinear Choquard equations in two dimensions under broad conditions on the nonlinearity.

Findings

Existence of groundstate solutions in or the nonlinear Choquard equation.

Applicable to general nonlinearities with growth and subcriticality conditions.

Provides a framework for analyzing similar equations in or mathematical physics.

Abstract

We prove the existence of a nontrivial groundstate solution for the class of nonlinear Choquard equation $$ -\Delta u+u=(I_\alpha*F(u))F'(u)\qquad\text{in }\mathbb{R}^2, $$ where $I_\alpha$ is the Riesz potential of order $\alpha$ on the plane $\mathbb{R}^2$ under general nontriviality, growth and subcriticality on the nonlinearity $F \in C^1 (\mathbb{R},\mathbb{R})$.