Normalized Schrödinger equations with mass-supercritical nonlinearity in exterior domains

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.