An equation involving the 1-Laplacian and a singular nonlinearity

We prove existence of solutions to a nonlinear degenerate elliptic equation of the form \ arrayll -Δ ₁ u+ |D u|(1-u)γ=g & in Ω ,\\ u=0 & on ∂ Ω , array. - Δ 1 u + | D u | ( 1 - u ) γ = g in Ω , u = 0 on ∂ Ω , in a suitable sense, where Ω Ω is a bounded open set of RN R N , γ >0 γ > 0 is a fixed parameter, g≥ 0 g ≥ 0 is a function in some Lebesgue space.