Liouville theorem for fully nonlinear elliptic equations with the small oscillation and the periodicity in $x$ and the periodic right hand term

Abstract

In this paper, we study quadratic growth solutions $u$ of fully nonlinear elliptic equations of the form $F(D^{2}u,x)=f$ 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 $F(D^{2}u)=f$ with the periodic data.

arXiv admin note: substantial text overlap with arXiv:2406.13927