The exact asymptotic behaviour of the unique solution to a singular nonlinear Dirichlet problem ✩

By Karamata regular varying theory, a perturbed argument and constructing comparison functions, we show the exact asymptotic behaviour of the unique solution u ∈ C2(Ω) ∩ C(Ω¯ ) near the boundary to a singular Dirichlet problem − u = b(x)g(u) + λf (u), u > 0, x ∈ Ω, u|∂Ω = 0, which is independent on λf (u), and we also show the existence and uniqueness of solutions to the problem, where Ω is a bounded domain with smooth boundary in RN, λ > 0, g ∈ C1((0,∞), (0,∞)) and there exists γ > 1 such that limt→0+ g (ξ t) g (t) = ξ−(1+γ ), ∀ξ >0, f ∈ Cα loc([0,∞), [0,∞)), the function f (s) s+s0 is decreasing on (0,∞) for some s0 > 0, and b is nonnegative nontrivial on Ω, which may be vanishing on the boundary. © 2006 Elsevier Inc. All rights reserved