Nonlocal Liouville theorems with gradient nonlinearity

TL;DR

This paper develops a unified approach using Ishii-Lions techniques to establish Liouville theorems for a broad class of nonlinear nonlocal equations with gradient nonlinearities, also addressing an open problem and exploring regularity applications.

Contribution

It introduces a new unified method for proving Liouville theorems in nonlinear nonlocal equations involving gradient terms, solving an open problem in the field.

Findings

Established Liouville properties for a large family of equations

Provided solutions to an open problem in the literature

Explored applications to regularity theory

Abstract

In this article we consider a large family of nonlinear nonlocal equations involving gradient nonlinearity and provide a unified approach, based on the Ishii-Lions type technique, to establish Liouville properties of the solutions. We also answer an open problem raised by [24]. Some applications to regularity issues are also studied.