  Let $\Omega\subset\mathbb{R}^N$ ($N\geq 3$) be a bounded $C^2$ domain and
$\Sigma\subset\partial\Omega$ be a compact $C^2$ submanifold of dimension $k$.
Denote the distance from $\Sigma$ by $d_\Sigma$. In this paper, we study
positive solutions of the equation $(*)\, -\Delta u -\mu u/d_\Sigma^2 =
g(u,|\nabla u|)$ in $\Omega$, where $\mu\leq \big( \frac{N-k}{2} \big)^2$ and
the source term $g:\mathbb{R}\times\mathbb{R}_+ \rightarrow \mathbb{R}_+$ is
continuous and non-decreasing in its arguments with $g(0,0)=0$. In particular,
we prove the existence of solutions of $(*)$ with boundary measure data $u=\nu$
in two main cases, provided that the total mass of $\nu$ is small. In the first
case $g$ satisfies some subcriticality conditions that always ensure the
existence of solutions. In the second case we examine power type nonlinearity
$g(u,|\nabla u|) = |u|^p|\nabla u|^q$, where the problem may not possess a
solution for exponents in the supercritical range. Nevertheless we obtain
criteria for existence under the assumption that $\nu$ is absolutely continuous
with respect to some appropriate capacity or the Bessel capacity of $\Sigma$,
or under other equivalent conditions.
