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

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.