Existence of classical solutions to the exterior Dirichlet problem for Hessian quotient equations

Abstract

This paper studies the exterior Dirichlet problem for Hessian quotient equations with nonconstant right-hand sides. We prove the existence of classical admissible solutions with prescribed asymptotic Hessians and establish convergence of the Hessian at infinity. The main difficulties are obtaining second-order estimates on expanding annuli that are uniform in the outer radius and deriving Hessian convergence under an integral tail condition with no prescribed decay rate. These are resolved through a radius-independent boundary-to-interior estimate and a blow-down argument. We also allow nonradial perturbations of the source. Under stronger pointwise assumptions, we obtain higher-order asymptotic expansions and solutions for every sufficiently large prescribed asymptotic constant.