The asymptotic behaviour of solutions with blow-up at the boundary for semilinear elliptic problems

By constructing the comparison functions and the perturbed method, it is showed that any solution u ∈ C2(Ω) to the semilinear elliptic problems Δu = k(x)g(u), x ∈ Ω, u|∂Ω =+∞ satisfies limd(x)→0 u(x) Z(dμ(x)) = (2+σ)(2+ρ+σ) 2c0(2+ρ) 1/ρ, where Ω is a bounded domain with smooth boundary in RN; limd(x)→0 k(x) dσ (x) = c0, −2 < σ, c0 > 0, μ = 2+σ 2 ; g ∈ C1[0,∞), g 0 and g(s) s is increasing on (0,∞), there exists ρ > 0 such that lims→∞ g (sξ ) g (s) = ξρ, ∀ξ > 0, ∞Z(s) dt √2G(t) = s, G(t) = t 0 g(s) ds.  2005 Elsevier Inc. All rights reserved