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 | − 2 a ∇ u ) = | x | − 2 ( a + 1 ) + c f ( x , u , ∇ u ) , in Ω u = 0 , on ∂ Ω , where the non-linearity f ( x , t , ξ ) belongs to a class of functions that are superlinear in the variable t and sublinear in the variable ξ . 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.