A Unified Boundary Behavior of Large Solutions to Hessian Equations

Abstract

This paper is concerned with strictly k-convex large solutions to Hessian equations S _k(D ₂ u(x)) = b(x)f(u(x)), x ∈ Ω, where Ω is a strictly (k − 1)-convex and bounded smooth domain in ℝ ⁿ, $b \in {C^\infty }\left( {\overline {\rm{\Omega }} } \right)$ is positive in Ω, but may be vanishing on the boundary. Under a new structure condition on f at infinity, the author studies the refined boundary behavior of such solutions. The results are obtained in a more general setting than those in [Huang, Y., Boundary asymptotical behavior of large solutions to Hessian equations, Pacific J. Math., 244, 2010, 85–98], where f is regularly varying at infinity with index p > k.

Keywords

Hessian equations / Strictly k-convex large solutions / Boundary behavior