Exterior Dirichlet Problems for Hessian Quotient Equations of Mixed Type

We study the exterior Dirichlet problem for the mixed Hessian quotient equation $$\frac{{\sigma }_k(\eta (D^{2}u))}{{\sigma }_l(\eta (D^{2}u))}=1,$$ 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.