Liouville property and existence of entire solutions of Hessian equations

TL;DR

This paper proves existence, uniqueness, and Liouville-type results for entire solutions of Hessian equations, extending prior work on Monge-Ampère equations and relaxing convexity restrictions.

Contribution

It establishes optimal asymptotic behavior for solutions and removes certain convexity restrictions, advancing the understanding of Hessian equations.

Findings

Proved existence and uniqueness of entire solutions with prescribed asymptotics.

Established a Liouville type theorem for $k$-convex solutions.

Extended results to broader classes of Hessian equations.

Abstract

In this paper, we establish the existence and uniqueness theorem for entire solutions of Hessian equations with prescribed asymptotic behavior at infinity. This extends the previous results on Monge-Amp\`{e}re equations. Our approach also makes the prescribed asymptotic order optimal within the range preset in exterior Dirichlet problems. In addition, we show a Liouville type result for $k$-convex solutions. This partly removes the $(k+1)$- or $n$-convexity restriction imposed in existing work.