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} -Δ_{1} u+ \frac{|D u|}{(1-u)^γ}=g & in $Ω$,\\ u=0 \hfill & on $\partialΩ$, \end{cases} in a suitable sense, where $\mathrm{\Omega }$ is a bounded open set of ${\mathbb{R}}^n$, $\gamma >0$ is a fixed parameter, $g\ge 0$ is a function in some Lebesgue space.