On the existence of positive solutions of singular nonlinear elliptic equations with Dirichlet boundary conditions

This paper is concerned with the existence of positive solutions of the singular nonlinear elliptic equation with a Dirichlet boundary condition{Δu=F(x,u)in Ω,u=ϕon ∂Ω, where F   is a Borel measurable function in Ω×(0,+∞)Ω×(0,+∞) such that |F(x,u)|⩽V(x)u−α|F(x,u)|⩽V(x)u−α for some α>0α>0 and V   satisfying some appropriate conditions. In particular, we show that the above problem has positive solutions whenever inf∂Ωϕinf∂Ωϕ is greater than a positive quantity given by α and V.