Positive normalized solutions of Schrödinger equations with Sobolev critical growth in bounded domains

This paper investigates the existence of positive normalized solutions to the Sobolev critical Schrödinger equation: \begin{equation*} \left{ \begin{aligned} &-Δu +λu =|u|^{2^-2}u \quad &\mbox{in}& \ Ω,\ &\int_Ω|u|^{2}dx=c, \quad u=0 \quad &\mbox{on}& \ \partialΩ, \end{aligned} \right. \end{equation} where $\mathrm{\Omega }\subset {\mathbb{R}}^N$ ($N\ge 3$) is a bounded smooth domain, $2^{\ast }=\frac{2N}{N-2}$, $\lambda \in \mathbb{R}$ is a Lagrange multiplier, and $c>0$ is a prescribed constant. By introducing a novel blow-up analysis for Sobolev subcritical approximation solutions with uniformly bounded Morse index and fixed mass, we establish the existence of mountain pass type positive normalized solutions for $N\ge 3$. This resolves an open problem posed in [Pierotti, Verzini and Yu, SIAM J. Math. Anal. 2025].