Normalized solutions for Choquard equations with critical nonlinearities on bounded domains

TL;DR

This paper establishes the existence of two positive normalized solutions for a nonlocal nonlinear Schrödinger equation with critical nonlinearities on bounded domains, including a ground state and a mountain pass solution.

Contribution

It introduces new results on normalized solutions for Choquard equations with critical nonlinearities, identifying both ground state and mountain pass solutions.

Findings

Existence of two positive normalized solutions.

One solution is a ground state, the other a mountain pass solution.

Solutions are for equations with nonlocal nonlinearities on bounded domains.

Abstract

The aim of this work is the study of the existence of normalized solutions to the nonlinear Schr\"odinger equation with nonlocal nonlinearities: \begin{equation}\nonumber \left\{\begin{aligned} &-\Delta u =\lambda u+(I_\alpha*|u|^{2_\alpha^*})|u|^{2_\alpha^*-2}u+a(I_\alpha*|u|^p)|u|^{p-2}u,\ x\in\Omega,\\ &u>0\ \text {in}\ \Omega,\ u=0\ \text {on}\ \partial \Omega,\ \int _{\Omega}|u|^2dx=c, \end{aligned} \right. \end{equation} where $c>0,\ \alpha \in (0,N),\ \frac{N+\alpha+2}{N}<p<\frac{N+\alpha}{N-2}=2_\alpha^*,\ a\ge 0,\ \Omega \subset \mathbb{R}^N (N \ge 3)$ is smooth, bounded, star-shaped and $I_\alpha$ is the Riesz potential. We prove the existence of two positive normalized solutions, one of which is a ground state and the other is a mountain pass solution.