Semi-linear singular elliptic equations with dependence on the gradient

Abstract

We establish the existence of a positive solution for the following non-variational equation {-div(|x|(-2a)del u)=|x|(-2(a+1)+c) f(x,u,del u), in Omega u=0, on partial derivative Omega,

where the non-linearity f (x,t,xi) belongs to a class of functions that are superlinear in the variable t and sublinear in the variable xi. 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.