  Let $\Omega$ be a smooth bounded domain in $\mathbb{R}^N$ and
$\delta(x)=\text{dist}\,(x,\partial \Omega)$. Assume $\mu>0$, $\nu$ is a
nonnegative finite measure on $\partial \Omega$ and $g \in C(\Omega \times
\mathbb{R}_+)$. We study positive solutions of $$ (P)\qquad -\Delta u -
\frac{\mu}{\delta^2} u = g(x,u) \text{ in } \Omega, \qquad \text{tr}^*(u)=\nu.
$$ Here $\text{tr}^*(u)$ denotes the normalized boundary trace of $u$ which was
recently introduced by M. Marcus and P. T. Nguyen. We focus on the case $0<\mu
< C_H(\Omega)$ (the Hardy constant for $\Omega$) and provide some qualitative
properties of solutions of (P). When $g(x,u)=u^q$ with $q>1$, we prove that
there is a critical value $q^*$ (depending only on $N$, $\mu$) for (P) in the
sense that if $1<q<q^*$ then (P) admits a solution under a smallness assumption
on $\nu$, but if $q \geq q^*$ this problem admits no solution with isolated
boundary singularity. Existence result is then extended to a more general
setting where $g$ is subcritical. We also investigate the case where the $g$ is
linear or sublinear and give some existence results for (P).
