A nonlinear Liouville theorem for fractional equations in the Heisenberg group

TL;DR

This paper proves a Liouville-type theorem for nonlinear fractional equations in the Heisenberg group, utilizing local realizations of fractional operators and geometric methods like CR inversion and moving planes.

Contribution

It introduces a novel Liouville theorem for fractional sub-Laplacian equations in the Heisenberg group using the Dirichlet-to-Neumann approach and geometric symmetry techniques.

Findings

Establishment of a Liouville-type theorem for fractional Heisenberg equations

Application of CR inversion and moving plane methods in the proof

Use of local realization of fractional CR covariant operators

Abstract

We establish a Liouville-type theorem for a subcritical nonlinear problem, involving a fractional power of the sub-Laplacian in the Heisenberg group. To prove our result we will use the local realization of fractional CR covariant operators, which can be constructed as the Dirichlet-to-Neumann operator of a degenerate elliptic equation in the spirit of Caffarelli and Silvestre, as established in \cite{FGMT}. The main tools in our proof are the CR inversion and the moving plane method, applied to the solution of the lifted problem in the half-space $\mathbb H^n\times \mathbbR^+$.