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 Schrodinger equation $$-\Delta u+\lambda u+V(x)u=|u|<^>{p-2}u,\quad u\in H<^>1_0(\Omega),\quad\int_\Omega u<^>2{\rm d}\,x=\rho<^>2,\quad\lambda\in\mathbb{R},$$ where $\Omega =\mathbb {R}<^>N$ or $\mathbb {R}<^>N\setminus \Omega$ 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?), or if $\|V\|_{L<^>q}$ is small, for some $q\ge \frac N2$ ($q>1$ if $N=2$). No smallness assumption on $V$ is required if the decay assumption (D?) is fulfilled. There are no assumptions on the size of $\mathbb {R}<^>N\setminus \Omega$. The solution $\bar u$ is a bound state and no ground state solution exists, up to the autonomous case $V\equiv 0$ and $\Omega =\mathbb {R}<^>N$.