  In this paper we prove existence of nonnegative bounded solutions for the
non-autonomous prescribed mean curvature problem in non-parametric form on an
open bounded domain $\Omega$ of $\mathbb{R}^N$. The mean curvature, that
depends on the location of the solution $u$ itself, is asked to be of the form
$f(x)h(u)$, where $f$ is a nonnegative function in $L^{N,\infty}(\Omega)$ and
$h:\mathbb{R}^+\mapsto \mathbb{R}^+$ is merely continuous and possibly
unbounded near zero. As a preparatory tool for our analysis we propose a purely
PDE approach to the prescribed mean curvature problem not depending on the
solution, i.e. $h\equiv 1$. This part, which has its own independent interest,
aims to represent a modern and up-to-date account on the subject. Uniqueness is
also handled in presence of a decreasing nonlinearity. The sharpness of the
results is highlighted by mean of explicit examples.
