Positive normalized solutions of Schr\"odinger equations with Sobolev critical growth in bounded domains

This paper investigates the existence of positive normalized solutions to the Sobolev critical Schr\"odinger equation: equation* \ aligned &- u + u =|u|^{2^*-2u &in& \, \\ &_|u|^2dx=c, u=0 &on& \, aligned. equation* where ^N (N3) is a bounded smooth domain, 2^*=2NN-2, 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 3. This resolves an open problem posed in Pierotti, Verzini and Yu, SIAM J. Math. Anal. 2025.