On a class of fully nonlinear elliptic equation in dimension two

TL;DR

This paper investigates the existence and asymptotic behavior of radial positive solutions to certain fully nonlinear elliptic equations involving Pucci's extremal operators in two dimensions, including a nonlinear Liouville equation.

Contribution

It establishes the existence of solutions for a nonlinear Liouville equation and identifies a critical exponent for the Pucci's operator in two dimensions.

Findings

Existence of positive solutions for the nonlinear Liouville equation.

Identification of a critical exponent for the Pucci's operator.

Bounds for the critical exponent are provided.

Abstract

We study existence and asymptotic behavior of radial positive solutions of some fully nonlinear equations involving Pucci's extremal operators in dimension two. In particular we prove the existence of a positive solution of a fully nonlinear version of the Liouville equation in the plane. Moreover for the $\mathcal{M}^-_{\lambda,\Lambda}$ operator, we show the existence of a critical exponent and give bounds for it.