On a quadratic gradient natural term for the Pucci extremal operators

TL;DR

This paper introduces a quadratic gradient term for Pucci extremal operators, extending classical concepts and analyzing solution properties under various conditions, with implications for nonlinear PDE theory.

Contribution

It proposes a new quadratic gradient term for Pucci operators and studies its effects on solution existence, uniqueness, and asymptotic behavior.

Findings

Extension of classical quadratic gradient term to Pucci operators

Analysis of solution existence and non-existence under new conditions

Investigation of asymptotic behavior and Liouville-type results

Abstract

We introduce a quadratic gradient type term for the Pucci extremal operators. Our analysis demonstrates that this proposed term extends the classical quadratic gradient term associated with the Laplace equation, and we investigate the impact of the Kazdan-Kramer transformation. As an application, we explore the existence, non-existence, uniqueness, Liouville-type results, and asymptotic behavior of solutions for the new class of Pucci equations under various conditions on both nonlinearity and domains.