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

Let $\mathrm{\Omega }\subset {\mathbb{R}}^N$ ($N\ge 3$) be a $C^2$ bounded domain and $F\subset \mathrm{\partial }\mathrm{\Omega }$ be a $C^2$ submanifold of dimension $0 \leq k \leq N-2$\mathrm{.}Put$\delta_F(x)=dist(x,F)$,$V=\delta_F^{-2}$in$\Omega$and$L_{\gamma V}=\Delta + \gamma V$\mathrm{.}Denoteby$C_H(V)$theHardyconstantrelativeto$V$in$\Omega$\mathrm{.}Westudypositivesolutionsofequations(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 $F=\mathrm{\partial }\mathrm{\Omega }$ - 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 $\mathrm{\Omega }$ and establish existence and stability results when the data is concentrated on the set of subcritical points.