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, whereΩ 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 ∈ Cα loc(Ω) for some α ∈ (0, 1), is nonnegative on Ω, which is also singular near the boundary.  2005 Elsevier Inc. All rights reserved