An equation involving the 1-Laplacian and a singular nonlinearity

Abstract

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