Boundary blow-up elliptic problems with nonlinear gradient terms

By Karamata regular variation theory and perturbation method, we show the exact asymptotical behaviour of solutions near the boundary to nonlinear elliptic problems Δu± uq=b(x)g(u), u>0 in Ω, u∂Ω=+∞, where Ω is a bounded domain with smooth boundary in , q 0, g C1[0,∞), g(0)=0, g′ is regularly varying at infinity with index ρ with ρ>0 and b is nonnegative nontrivial in Ω, which may be vanishing on the boundary