Normalized positive solutions for Schrödinger equations with potentials in unbounded domains

The paper deals with the existence of positive solutions with prescribed $L^2$ norm for the Schrödinger equation $$-\mathrm{\Delta }u+\lambda u+V(x)u=\mathrm{\mid }u{\mathrm{\mid }}^{p-2}u,u\in H_0^1(\mathrm{\Omega }),{\int }_{\mathrm{\Omega }}{u}^{2}dx={\rho }^2,\lambda \in \mathbb{R},$$ where $\mathrm{\Omega }={\mathbb{R}}^N$ or ${\mathbb{R}}^N\setminus \mathrm{\Omega }$ is a compact set, $\rho >0$, $V\ge 0$ (also $V\equiv 0$ is allowed), $p\in (2,2+\frac{4}{N})$. The existence of a positive solution $\bar{u}$ is proved when $V$ verifies a suitable decay assumption $(D_{\rho })$, or if $\mathrm{\parallel }V{\mathrm{\parallel }}_{L^q}$ is small, for some $q\ge \frac{N}{2}$ ($q>1$ if $N=2$). No smallness assumption on $V$ is required if the decay assumption $(D_{\rho })$ is fulfilled. There are no assumptions on the size of ${\mathbb{R}}^N\setminus \mathrm{\Omega }$. The solution $\bar{u}$ is a bound state and no ground state solution exists, up to the autonomous case $V\equiv 0$ and $\mathrm{\Omega }={\mathbb{R}}^N$.