Positive normalized solutions of Schrödinger equations with Sobolev critical growth in bounded domains
Xiaojun Chang, Manting Liu, Duokui Yan
TL;DR
We address the existence of a second positive normalized solution for the Sobolev-critical Schrödinger equation $-\ abla u^2 + \lambda u = |u|^{2^*-2}u$ in a bounded domain with mass constraint $\int_\Omega |u|^2 dx = c$. The authors introduce a Sobolev subcritical approximation with exponents $p\in(2,2^*)$, coupled with a novel blow-up analysis on bounded domains and uniform Morse-index control, to overcome compactness issues at the critical exponent. They prove that for $N\ge 3$ there exists $c^{**}>0$ such that, for all $0<c<c^{**}$, a second positive normalized solution of mountain-pass type exists—distinct from the local minimizer—by comparing the mountain-pass energy with the local-minimizer energy and establishing strong convergence of subcritical approximants. This work advances normalized-solution theory under critical growth in bounded domains and introduces analytic tools (a new blow-up function and profile decomposition) that may apply to related constrained critical problems.
Abstract
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 $Ω\subset\mathbb{R}^{N}$ ($N\geq3$) is a bounded smooth domain, $2^*=\frac{2N}{N-2}$, $λ\in \mathbb{R}$ 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].