Complete Classification of the Euclidean Complete Solutions to a Monge-Ampere Equation

TL;DR

This paper classifies all Euclidean complete solutions to a Monge-Ampère equation with power terms, providing sharp conditions on parameters and domains.

Contribution

It offers a complete classification of Euclidean complete solutions to a Monge-Ampère equation with explicit conditions on p and domain.

Findings

Provides sharp conditions on p and domain for Euclidean completeness.

Classifies all Euclidean complete solutions explicitly.

Establishes criteria for solutions defined over entire space or large hypersurfaces.

Abstract

We study a Monge-Amp\`{e}re equation with power term for some $p\in{\mathbb{R}}$. A solution $u$ is called to be Euclidean complete if it is an entire solution defined over the whole ${\mathbb{R}}^n$ or its graph is a large hypersurface satisfying the large condition $u(x)\to\infty$ as $\mathrm{dist}(x,\partial\Omega)\to 0$ in case of $\Omega\not={\mathbb{R}}^n$. In this paper, we will give various sharp conditions on $p$ and $\Omega$ classifying the Euclidean complete solutions.