By Karamata regular variation theory, a perturbation method and construction of comparison functions, we show the exact asymptotic behaviour of solutions near the boundary to nonlinear elliptic problems Δu ± |Δu|q = b(x)g(u), u > 0 in Ω, u|∂Ω = ∞, where Ω is a bounded domain with smooth boundary in ℝN, q > 0, g ∈ C1[0, ∞) is increasing on [0, ∞), g(0) = 0, g′ is regularly varying at infinity with positive index ρ and b is non-negative in Ω and is singular on the boundary.