A second-order estimate for blow-up solutions of elliptic equations

We investigate second-term asymptotic behavior of boundary blow-up solutions to the problems Δu=b(x)f(u)Δu=b(x)f(u), x∈Ωx∈Ω, subject to the singular boundary condition u(x)=∞u(x)=∞, in a bounded smooth domain Ω⊂RNΩ⊂RN. b(x)b(x) is a non-negative weight function. The nonlinearly ff is regularly varying at infinity with index ρ>1ρ>1 (that is limu→∞f(ξu)/f(u)=ξρlimu→∞f(ξu)/f(u)=ξρ for every ξ>0ξ>0) and the mapping f(u)/uf(u)/u is increasing on (0,+∞)(0,+∞). The main results show how the mean curvature of the boundary ∂Ω∂Ω appears in the asymptotic expansion of the solution u(x)u(x). Our analysis relies on suitable upper and lower solutions and the Karamata regular variation theory.