Liouville type theorems for fully nonlinear elliptic equations in exterior domains in half spaces

We establish a Liouville type theorem for fully nonlinear uniformly elliptic equations in exterior domains in half spaces under quadratic boundary data and a quadratic growth condition, that is, any viscosity solution tends to a quadratic polynomial plus lower order terms at infinity with the rate at least $\mathrm{\mid }x{\mathrm{\mid }}^{-\alpha -n}$ for any $\alpha \in (0,1)$. As applications, this result can lead to Liouville type theorems for Monge-Amp`ere equations, k-Hessian equations and special Lagrangian equations with critical and supercritical phases.