Quadratic growth solutions of fully nonlinear elliptic equations with periodic data

TL;DR

This paper investigates quadratic growth solutions to fully nonlinear elliptic equations with periodic data, establishing existence and Liouville theorems that generalize classical results, including applications to k-Hessian and Monge-Ampère equations.

Contribution

It introduces new existence and Liouville theorems for quadratic growth solutions with periodic data, extending classical results to non-uniformly elliptic cases.

Findings

Established existence of solutions in whole space and exterior domains.

Proved Liouville type theorems for solutions with quadratic growth.

Applied results to k-Hessian and Monge-Ampère equations.

Abstract

In this paper, we study quadratic growth solutions $u$ of fully nonlinear elliptic equations of the form $F(D^2u)=f$ in $\mathbb{R}^n$, where $f$ is periodic and $F$ may be not uniformly elliptic. The existence of solutions and Liouville type results in the whole space and exterior domains are established, which generalize the classical results when $f$ is constant. As applications, the corresponding results are given to $k$-Hessian equations, which include the celebrated results for Monge-Amp\`{e}re equations.