This paper investigates the existence of positive normalized solutions to the Sobolev critical Schr\"{ o}dinger equation: \begin{ equation*} \left\{ \begin{ aligned} &-\Delta u +\lambda u =|u|^{ 2^*-2}u \quad &\mbox{ in}& \ \Omega,\\ &\int_{ \Omega}|u|^{ 2}dx=c, \quad u=0 \quad &\mbox{ on}& \ \partial\Omega, \end{ aligned} \right. \end{ equation*} where $\Omega\subset\mathbb{ R}^{ N}$ ($N\geq3$) is a bounded smooth domain, $2^*=\frac{ 2N}{ N-2}$, $\lambda\in \mathbb{ R}$ 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].
Categories: math.AP