Towards a Liouville theorem for continuous viscosity solutions to fully nonlinear elliptic equations in conformal geometry

TL;DR

This paper establishes a Liouville theorem for continuous viscosity solutions to fully nonlinear elliptic equations in conformal geometry, showing such solutions are either constants or standard bubbles under certain conditions.

Contribution

It proves the strong comparison principle and Hopf Lemma for non-uniformly elliptic equations involving the conformal Hessian, leading to a Liouville theorem for entire solutions.

Findings

Proved strong comparison principle and Hopf Lemma for certain elliptic equations.

Established Liouville theorem classifying entire solutions as constants or bubbles.

Applied results to solutions approximable by $C^{1,1}$ functions on larger domains.

Abstract

We study entire continuous viscosity solutions to fully nonlinear elliptic equations involving the conformal Hessian. We prove the strong comparison principle and Hopf Lemma for (non-uniformly) elliptic equations when one of the competitors is $C^{1,1}$. We obtain as a consequence a Liouville theorem for entire solutions which are approximable by $C^{1,1}$ solutions on larger and larger compact domains, and, in particular, for entire $C^{1,1}_{\rm loc}$ solutions: they are either constants or standard bubbles.