Fully Nonlinear Elliptic Equations With Periodic Data

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.