  In this paper, under very general assumptions, we prove existence and
regularity of distributional solutions to homogeneous Dirichlet problems of the
form $$\begin{cases} \displaystyle - \Delta_{1} u = h(u)f & \text{in}\,
\Omega,\newline u\geq 0& \text{in}\ \Omega, \newline u=0 & \text{on}\ \partial
\Omega, \end{cases} $$ where, $\Delta_{1} $ is the $1$-laplace operator,
$\Omega$ is a bounded open subset of $\mathbb{R}^N$ with Lipschitz boundary,
$h(s)$ is a continuous function which may become singular at $s=0^{+}$, and $f$
is a nonnegative datum in $L^{N,\infty}(\Omega)$ with suitable small norm.
Uniqueness of solutions is also shown provided $h$ is decreasing and $f>0$. As
a by-product of our method a general theory for the same problem involving the
$p$-laplacian as principal part, which is missed in the literature, is
established. The main assumptions we use are also further discussed in order to
show their optimality.

Analysis of PDEs