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.