The asymptotic behaviour of the unique solution for the singular Lane-Emden-Fowler equation

By Karamata regular variation theory and constructing comparison functions, we show the exact asymptotic behaviour of the unique classical solution u∈C2(Ω)∩C(Ω¯) near the boundary to a singular Dirichlet problem −Δu=k(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,∞)), limt→0+g(ξt)g(t)=ξ−γ, for each ξ>0, for some γ>0; and k∈Clocα(Ω) for some α∈(0,1), is nonnegative on Ω, which is also singular near the boundary.