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)→0u(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)→0k(x)dσ(x)=c0, −2<σ, c0>0, μ=2+σ2; 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)∞dt2G(t)=s, G(t)=∫0tg(s)ds.