Positive normalized solutions of Schr\"{o}dinger equations with Sobolev critical growth in bounded domains

This paper investigates the existence of positive normalized solutions to the Sobolev critical Schr\"{o}dinger equation: {equation*} \{ {aligned} &-Δ u +λ u =|u|2^*-2u &in& \ Ω,\\ &∫Ω|u|²dx=c, u=0 &on& \ ∂Ω, {aligned} . {equation*} where ΩN (N≥3) is a bounded smooth domain, 2^*=2N/N-2, λ∈ 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].