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

TL;DR

This paper proves the existence of positive normalized solutions for Sobolev critical Schrödinger equations in bounded domains using a new blow-up analysis, addressing an open problem in the field.

Contribution

It introduces a novel blow-up analysis method for Sobolev subcritical approximations with bounded Morse index, establishing solutions for the critical case.

Findings

Existence of mountain pass type solutions for N≥3.

Resolution of an open problem in the literature.

Development of a new analytical technique for Sobolev critical problems.

Abstract

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].