On the exterior Dirichlet problem for Hessian quotient equations

TL;DR

This paper proves existence and uniqueness for solutions to the exterior Dirichlet problem for Hessian quotient equations, extending previous results on related equations and employing Perron's method with new subsolution constructions.

Contribution

It introduces a systematic approach to solve the exterior Dirichlet problem for Hessian quotient equations using novel subsolution techniques and extends prior work on Monge-Ampère and Hessian equations.

Findings

Established existence and uniqueness of solutions

Constructed appropriate subsolutions using elementary symmetric functions

Extended results to more general Hessian quotient equations

Abstract

In this paper, we establish the existence and uniqueness theorem for solutions of the exterior Dirichlet problem for Hessian quotient equations with prescribed asymptotic behavior at infinity. This extends the previous related results on the Monge-Amp\`{e}re equations and on the Hessian equations, and rearranges them in a systematic way. Based on the Perron's method, the main ingredient of this paper is to construct some appropriate subsolutions of the Hessian quotient equation, which is realized by introducing some new quantities about the elementary symmetric functions and using them to analyze the corresponding ordinary differential equation related to the generalized radially symmetric subsolutions of the original equation.