Normalized Schrödinger equations with mass-supercritical nonlinearity in exterior domains

We consider the problem $-\mathrm{\Delta }u+\lambda u=u^{p-1}$, where $u\in H_0^1(\mathrm{\Omega })$ verifies $\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}=m>0$, and $\lambda \in [0,+\mathrm{\infty })$. Here, ${\mathbb{R}}^N\setminus \mathrm{\Omega }$ 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 ${\mathbb{R}}^N\setminus \mathrm{\Omega }$ in a small ball and in the case $\mathrm{\Omega }$ fixed and $m$ large.