Normalized solutions for Choquard equations with critical 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.