  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.
