Optimal asymptotic expansion of entire solutions to Monge-Amp\`{e}re equation with $C^\alpha$ perturbed periodic data

Abstract

We consider the asymptotic behavior at infinity of solution $u$ to Monge-Amp\`{e}re equation $\det(D^2u)=f$ in $\rn$, where $f$ is a perturbation of a periodic function and is only assumed to be H\"{o}lder continuous, compared to the previous work that $f$ is at least $C^{1,\az}$. The consequence established in this paper, by a nonlocal method, is that the difference between $u$ and a quadratic polynomial is asymptotically close to a periodic function.