Solvability of Hessian quotient equations in exterior domains

In this paper, we study the Dirichlet problem of Hessian quotient equations of the form Sk (D2u)/Sl (D2u) = g(x) in exterior domains. For g ≡ const., we obtain the necessary and sufficient conditions on the existence of radially symmetric solutions. For g being a perturbation of a generalized symmetric function at infinity, we obtain the existence of viscosity solutions by Perron's method. The key technique we develop is the construction of sub- and supersolutions to deal with the non-constant right-hand side g.