Quadratic growth solutions of fully nonlinear elliptic equations with periodic data

Abstract

In this paper, we study quadratic growth solutions $u$ of fully nonlinear elliptic equations of the form $F(D^2u)=f$ in $\mathbb{R}^n$, where $f$ is periodic and $F$ may be not uniformly elliptic. The existence of solutions and Liouville type results in the whole space and exterior domains are established, which generalize the classical results when $f$ is constant. As applications, the corresponding results are given to $k$-Hessian equations, which include the celebrated results for Monge-Amp\`{e}re equations.