Schr\"odinger equations with singular potentials: linear and nonlinear boundary value problems

TL;DR

This paper investigates boundary value problems for Schr"odinger equations with singular potentials, establishing boundary trace concepts, a priori estimates, and existence results for both linear and nonlinear cases in bounded domains.

Contribution

It introduces a normalized boundary trace for positive solutions, analyzes subsolutions and supersolutions at the boundary, and explores subcriticality and stability for nonlinear equations with singular potentials.

Findings

Existence of normalized boundary trace for solutions of linear equations.

Boundary behavior analysis of subsolutions and supersolutions.

Conditions for existence and stability of solutions with concentrated data.

Abstract

Let $\Omega \subset {\mathbb R}^N$ ($N \geq 3$) be a $C^2$ bounded domain and $F \subset \partial \Omega$ be a $C^2$ submanifold of dimension $0 \leq k \leq N-2$. Put $\delta_F(x)=dist(x,F)$, $V=\delta_F^{-2}$ in $\Omega$ and $L_{\gamma V}=\Delta + \gamma V$. Denote by $C_H(V)$ the Hardy constant relative to $V$ in $\Omega$. We study positive solutions of equations (LE) $-L_{\gamma V} u = 0$ and (NE) $-L_{\gamma V} u+ f(u) = 0$ in $\Omega$ when $\gamma < C_H(V)$ and $f \in C({\mathbb R})$ is an odd, monotone increasing function. We establish the existence of a normalized boundary trace for positive solutions of (LE) - first studied by Marcus and Nguyen for the case $F=\partial \Omega$ - and employ it to investigate the behavior of subsolutions and super solutions of (LE) at the boundary. Using these results we study boundary value problems for (NE) and derive a-priori estimates. Finally we discuss subcriticality of (NE) at boundary points of $\Omega$ and establish existence and stability results when the data is concentrated on the set of subcritical points.