The existence and asymptotic behavior of large solutions to the k -Hessian equations with nonlinear gradient terms

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 . ${S}_{k}\left({D}^{2}u\right)=b\left(x\right)f\left(u\right){\left({C}_{0}+{\left\vert \nabla u\right\vert }^{2}\right)}^{q/2}\,\text{in}\,{\Omega}, u=+\infty \,\text{on}\,\partial {\Omega}.$ Here, Ω is a smooth, bounded, and strictly convex domain in R N ( N ≥ 2 ) ${\mathbb{R}}^{N} \left(N\ge 2\right)$ , 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 ) $f\in {C}^{\infty }\left(\mathbb{R}\right)$ ) is positive and increasing on [0, + ∞ ) (or R $\mathbb{R}$ ). 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