Normalized Schrödinger equations with mass-supercritical nonlinearity in exterior domains
Luigi Appolloni, Riccardo Molle
TL;DR
This work studies normalized solutions to the mass-constrained Schrödinger equation $-\Delta u+\lambda u=|u|^{p-2}u$ on exterior domains with mass $|u|_2=m$ in the mass-supercritical range $p\in(2+4/N,2N/(N-2))$. Lacking a Pohozaev-based tool in exterior domains, the authors develop an $\eta$-approximating family and a novel linking structure to produce bounded Palais–Smale sequences with constrained Morse index $\widetilde{i}\le N+1$, and a positive Lagrange multiplier. Through a careful weak-limit and blow-up analysis, they show convergence to a nontrivial constrained solution for $\eta\nearrow 1$, establishing existence of a positive solution with $\widetilde{i}(u)\le N+1$ in two regimes: small exterior complement (for fixed $m$) and large mass (for fixed $\Omega$). The results provide, for the first time in exterior domains, normalized solutions in the mass-supercritical setting and rely on a refined variational approach that avoids Pohozaev-type identities. The approach yields insights into concentration phenomena and energy geometry in unbounded domains with compact complement.
Abstract
We consider the problem $-Δu+λu=u^{p-1}$, where $u\in H^1_0(Ω)$ verifies $\|u\|_{L^2}=m>0$, and $λ\in [0,+\infty)$. Here, $\mathbb{R}^N\setminusΩ$ is nonempty and compact. We prove the existence of a solution with a constrained Morse index lower than or equal to $N+1$, both in the case $m$ fixed and $\mathbb{R}^N\setminusΩ$ in a small ball and in the case $Ω$ fixed and $m$ large.