Existence of normalized solutions to Choquard equation with general mixed nonlinearities

TL;DR

This paper investigates the existence of normalized solutions to a Choquard equation with mixed nonlinearities, covering subcritical, supercritical, and critical cases, using variational methods and critical point theory.

Contribution

It establishes the existence of solutions for a broad class of nonlinearities in the Choquard equation, including mixed and critical cases, extending previous results.

Findings

Existence of solutions in subcritical and supercritical regimes.

Results cover Hardy-Littlewood-Sobolev critical cases.

Solutions are characterized via variational methods.

Abstract

We study the existence of normalized solutions to the following Choquard equation with $F$ being a Berestycki-Lions type function \begin{equation*} \begin{cases} -\Delta u+\lambda u=(I_{\alpha}\ast F(u))f(u),\quad \text{in}\ \mathbb{R}^N, \\ \int_{\mathbb{R}^N}|u|^2dx=\rho^2, \end{cases} \end{equation*} where $N\geq 3$, $\rho>0$ is assigned, $\alpha\in (0,N)$, $I_{\alpha}$ is the Riesz potential, and $\lambda\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier. Here, the general nonlinearity $F$ contains the $L^2$-subcritical and $L^2$-supercritical mixed case, the Hardy-Littlewood-Sobolev lower critical and upper critical cases.