Second order expansion for the solution to a singular Dirichlet problem

摘要

In this paper, we analyze the second order expansion for the unique solution near the boundary to the singular Dirichlet problem −▵u=b(x)g(u),u>0,x∈Ω,u|∂Ω=0, where Ω is a bounded domain with smooth boundary in RN, g ∈ C1((0, ∞), (0, ∞)), g is decreasing on (0, ∞) with lims→0+g(s)=∞ and g is normalized regularly varying at zero with index −γ (γ > 1), b∈Clocα(Ω) (0 < α < 1), is positive in Ω, may be vanishing or singular on the boundary and belongs to the Kato class K(Ω). Our analysis is based on the sub-supersolution method and Karamata regular variation theory.