Abstract In this paper, we establish the first- and second-order asymptotic behaviors (expansions) of boundary blow-up solutions to the k -Hessian problem S k (D 2 u) = b (x) f (u) C 0 + ∇ u 2 q / 2 in Ω, u = + ∞ on ∂ Ω. S₊ (D^2u) =b (x) f (u) ({C₀+ u ^2) }^q/2\, in\, , u=+ \, on\,. Here, Ω is a smooth, bounded, and strictly convex domain in R N (N ≥ 2) R^N (N 2), q ∈ [0, k + 1), C 0 > 0 is a constant, b ∈ C ∞ (Ω) is positive in Ω and may be (critically) singular or vanish on ∂ Ω, and f ∈ C ∞ (0, + ∞) ∩ C [0, + ∞) (or f ∈ C ∞ (R) f C^ (R) ) is positive and increasing on [0, + ∞) (or R R). Under various conditions, we reveal the influences of principal curvatures of ∂ Ω and the gradient term on the coefficients of the expansions of large solutions to this problem. Particularly, when b becomes critically singular on
Haitao Wan (Thu,) studied this question.