Semi-linear singular elliptic equations with dependence on the gradient

We establish the existence of a positive solution for the following non-variational equation {−div(|x|−2a∇u)=|x|−2(a+1)+cf(x,u,∇u),in Ωu=0,on ∂Ω, where the non-linearity f(x,t,ξ)f(x,t,ξ) belongs to a class of functions that are superlinear in the variable tt and sublinear in the variable ξξ. For this purpose we used an idea of a recent work by De Figueiredo et al. [D. De Figueiredo, M. Girardi, M. Matzeu, Semilinear elliptic equations with dependence on the gradient via mountain-pass techniques, Diff. Integral Equ. (in press)] and we established a new regularity result for a class of Singular Elliptic Equations.