Normalized solutions for the Choquard equation with mass supercritical nonlinearity

TL;DR

This paper establishes the existence and multiplicity of normalized solutions for the nonlinear Choquard equation with mass supercritical nonlinearity, using variational methods under general conditions on the nonlinearity.

Contribution

It introduces new results on normalized solutions for the Choquard equation with supercritical nonlinearity, expanding understanding of solution multiplicity.

Findings

Existence of normalized solutions under broad conditions.

Multiple solutions are proven to exist.

Results applicable to a range of nonlinearities.

Abstract

We consider the nonlinear Choquard equation $$\begin{cases} & - \Delta u = (I_\alpha \ast F(u))F'(u) -\mu u \ \text{in}\ \mathbb{R}^N, & u \in \ H^1(\mathbb{R}^N), \ \int_{\mathbb{R}^N} |u|^2 dx=m, \end{cases} $$ where $\alpha\in(0,N)$, $m>0$ is prescribed, $\mu \in \mathbb{R}$ is a Lagarange multiplier, and $I_\alpha$ is the Riesz potential. Under general assumptions on the nonlinearity $F,$ we prove the existence and multiplicity of normalized solutions.