  In this paper we study existence and uniqueness of solutions to Dirichlet
problems as $$
  \begin{cases}
  g(u) -{\rm div}\left(\frac{D u}{\sqrt{1+|D u|^2}}\right) = f &
\text{in}\;\Omega,\\ \newline
  u=0 & \text{on}\;\partial\Omega,
  \end{cases} $$ where $\Omega$ is an open bounded subset of $\mathbb{R}^N$
($N\geq 2$) with Lipschitz boundary, $g:\mathbb{R}\to\mathbb{R}$ is a
continuous function and $f$ belongs to some Lebesgue spaces. In particular,
under suitable saturation and sign assumptions, we explore the regularizing
effect given by the absorption term $g(u)$ in order to get a solutions for data
$f$ merely belonging to $L^1(\Omega)$ and with no smallness assumptions on the
norm. We also prove a sharp boundedness result for data in $L^{N}(\Omega)$ as
well as uniqueness if $g$ is increasing.
