Exterior Dirichlet Problems for Hessian Quotient Equations of Mixed Type

Abstract

We study the exterior Dirichlet problem for the mixed Hessian quotient equation \[ \frac{σ_k(η(D^2 u))}{σ_l(η(D^2 u))} = 1, \] where $\eta (M)=(\mathrm{tr}M)I-M$. We establish existence and uniqueness of smooth admissible solutions with prescribed quadratic asymptotics at infinity, and obtain full derivative decay of the remainder. The proof relies on a three-stage subsolution construction.