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 $$ {\left\{ \begin{array}{ll} -\Delta _{1} u+ \frac{|D u|}{(1-u)^{\gamma }}=g & \hbox {in } \Omega ,\\ u=0 & \text{ on } \partial \Omega , \end{array}\right. } $$ - Δ 1 u + | D u | ( 1 - u ) γ = g in Ω , u = 0 on ∂ Ω , in a suitable sense, where $$\Omega $$ Ω is a bounded open set of $$\mathbb {R}^{N}$$ R N , $$\gamma >0$$ γ > 0 is a fixed parameter, $$g\ge 0 $$ g ≥ 0 is a function in some Lebesgue space.