Normalized positive solutions for Schrödinger equations with potentials in unbounded domains

Abstract

The paper deals with the existence of positive solutions with prescribed $L^2$ norm for the Schrödinger 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 (Dρ), 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 (Dρ) 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 $\Omega =\mathbb {R}^N$ .