An equation involving the 1-Laplacian and a singular nonlinearity

We prove existence of solutions to a nonlinear degenerate elliptic equation of the form \begin{cases} -\Delta_{1} u+ \frac{|D u|}{(1-u)^{\gamma}}=g & \mbox{in $\Omega$,}\\ u=0 \hfill & \mbox{on $\partial\Omega$,} \end{cases} in a suitable sense, where $\Omega$ is a bounded open set of $\mathbb{R}^n$, $\gamma>0$ is a fixed parameter, $g\geq 0$ is a function in some Lebesgue space.