  In this paper, we study solutions $u$ of fully nonlinear elliptic equations
of the form $F(D^2u)=f$ in $\mathbb{R}^n$, where $f$ is periodic. We establish
the existence and Liouville type results for entire quadratic polynomial growth
solutions, that is, the solution is a quadratic polynomial plus a periodic
function. As a consequence, we consider applications to $k$-Hessian equations.
