In this paper, we study quadratic growth solutions $u$ of fully nonlinear elliptic equations of the form $F(D^2u,x)=f$ in $\mathbb{ R}^n$, where $f$ is periodic and $F$ has the periodicity in $x$. Under the assumption that the oscillation of $F(M,x)$ in $x$ is ``small", we establish the existence and Liouville type results for quadratic growth solutions, which can be expressed into the sum of a quadratic polynomial and a periodic function. Consequently, these results are generalization of the existing results for linear elliptic equations $a_{ ij}D_{ ij}u=0$ and fully nonlinear elliptic equations $F(D^2u)=f$ with the periodic data.
Categories: math.AP