The Dirichlet problem for Hessian quotient equations on exterior domains

TL;DR

This paper proves existence and uniqueness of viscosity solutions for the exterior Dirichlet problem involving Hessian quotient equations with specific asymptotic behavior, extending prior results for related equations.

Contribution

It introduces a new existence and uniqueness theorem for Hessian quotient equations on exterior domains with prescribed asymptotics, generalizing previous work on Monge-Ampère and Hessian equations.

Findings

Established existence of viscosity solutions under specified conditions.

Proved uniqueness using comparison principles and Perron's method.

Extended previous results to a broader class of Hessian quotient equations.

Abstract

In this paper, we consider the exterior Dirichlet problem for Hessian quotient equations with the right hand side $g$, where $g$ is a positive function and $g=1+O(|x|^{-\beta})$ near infinity, for some $\beta>2$. Under a prescribed generalized symmetric asymptotic behavior at infinity, we establish an existence and uniqueness theorem for viscosity solutions, by using comparison principles and Perron's method. This extends the previous results for Monge--Amp\`ere equations and Hessian equations.