The second order estimate for the solution to a singular elliptic boundary value problem

Abstract

We study the second order estimate for the unique solution near the boundary to the singular Dirichlet problem -Δu=b(x)g(u), u > 0, x ∈ Ω, u| Ω= 0, where Ω is a bounded domain with smooth boundary in R N, g ∈ C 1((0,∞), (0,∞)), g is decreasing on (0,∞) with lim s→0+g(s) = ∞ and g is normalized regularly varying at zero with index -γ(γ> 1), b ∈ C α(Ω̄)(0 < α< 1), is positive in Ω, may be vanishing on the boundary. Our analysis is based on Karamata regular variation theory.