Schrödinger equations with singular potentials: linear and nonlinear boundary value problems

Abstract

Let $\Omega \subset \mathbb {R}^N$ ($N \ge 3$) be a $C^2$ bounded domain and $F \subset \partial \Omega$ be a $C^2$ submanifold with dimension $0 \le k \le N-2$. Denote $\delta _F=\mathrm {dist}\,(\cdot ,F)$, $V=\delta _F^{-2}$ and $C_H(V)$ 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 $\Omega$ where $L_{\gamma V}=\Delta + \gamma V$, $\gamma < C_H(V)$ and $f \in C(\mathbb {R})$ is an odd, monotone increasing function. We extend the notion of normalized boundary trace introduced in Marcus and Nguyen (Ann Inst H. Poincaré (C) Non Linear Anal 34:69–88, 2015) and employ it to investigate the linear equation (LE). Using these results we obtain properties of moderate solutions of (NE). Finally we determine a criterion for subcriticality of points on $\partial {\Omega }$ relative to f and study b.v.p. for (NE). In particular we establish existence and stability results when the data is concentrated on the set of subcritical points.