Normalized Schrödinger equations with mass-supercritical nonlinearity in exterior domains

Abstract

We consider the problem $-\hat{I}”u+\hat{I}»u=u^{p-1}$, where $u\in H_0^1(\hat{I}\text{c◯})$ verifies $\Vert u{\Vert }_{L^2}=m>0$, and $\hat{I}»\in [0,+\infty )$. Here, ${\mathbb{R}}^N\setminus \hat{I}\text{c◯}$ 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 ${\mathbb{R}}^N\setminus \hat{I}\text{c◯}$ in a small ball and in the case $\hat{I}\text{c◯}$ fixed and $m$ large.

28 pages