Singular quasilinear equations with quadratic growth in the gradient without sign condition

Given a bounded domain Ω in R N , and a function a ∈ L q ( Ω ) with q > N / 2 , we study the existence of a positive solution for the quasilinear problem − Δ w + g ( x , w ) | ∇ w | 2 = a ( x ) , x ∈ Ω , w ∈ H 0 1 ( Ω ) , where g ( x , s ) is a Carathéodory function on Ω × ( 0 , + ∞ ) which may have a singularity at s = 0 and may change of sign.