The exterior Dirichlet problems of Hessian quotient equations

TL;DR

This paper investigates the existence of solutions to Hessian quotient equations in exterior domains, establishing conditions for radial solutions and constructing viscosity solutions using ODE methods and Perron's technique.

Contribution

It provides new necessary and sufficient conditions for the existence of radial solutions and develops a method to construct viscosity solutions for exterior Hessian quotient problems.

Findings

Established conditions for the existence of radial solutions.

Constructed viscosity subsolutions and supersolutions using ODE techniques.

Proved the existence of viscosity solutions via Perron's method.

Abstract

In this paper, we study the Dirichlet problem of Hessian quotient equations in exterior domains. By estimating the eigenvalues of the solution, the necessary and sufficient conditions on existence of radial solutions are obtained. Applying the solutions of ODE, the viscosity subsolutions and supersolutions are constructed and then the existence of viscosity solutions for exterior problems is established by the Perron's method.