Normalized positive solutions for Schr\"odinger equations with potentials in unbounded domains

TL;DR

This paper establishes the existence of positive solutions with prescribed $L^2$ norm for Schrödinger equations in unbounded domains under decay or smallness conditions on the potential, without size restrictions on the domain complement.

Contribution

It proves existence of bound state solutions for Schrödinger equations with potentials satisfying decay or smallness conditions, extending results to unbounded domains without size restrictions.

Findings

Existence of positive solutions under decay condition $(D_ ho)$

Existence under small $L^q$ norm of $V$

No ground state solutions in the autonomous case

Abstract

The paper deals with the existence of positive solutions with prescribed $L^2$ norm for the Schr\"odinger equation $$ -\Delta u+\lambda u+V(x)u=|u|^{p-2}u,\qquad u\in H^1_0(\Omega),\quad\int_\Omega u^2dx=\rho^2,\quad\lambda\in\mathbb{R}, $$ 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), $p\in \left(2,2+\frac 4 N\right)$. The existence of a positive solution $\bar u$ is proved when $V$ verifies a suitable decay assumption $(D_\rho)$, 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 $(D_\rho)$ 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 $\Omega=\mathbb{R}^N$.