Normalized solutions for Choquard equations with critical nonlinearities on bounded domains

The aim of this work is the study of the existence of normalized solutions to the nonlinear Schrödinger equation with nonlocal nonlinearities: \begin{equation}\nonumber \left\{\begin{aligned} &-Δu =λu+(I_α*|u|^{2_α^*})|u|^{2_α^*-2}u+a(I_α*|u|^p)|u|^{p-2}u,\ x\inΩ,\\ &u>0\ \text {in}\ Ω,\ u=0\ \text {on}\ \partial Ω,\ \int _Ω|u|^2dx=c, \end{aligned} \right. \end{equation} where $c>0,\ α\in (0,N),\ \frac{N+α+2}{N}<p<\frac{N+α}{N-2}=2_α^*,\ a\ge 0,\ Ω\subset \mathbb{R}^N (N \ge 3)$ is smooth, bounded, star-shaped and $I_α$ 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.