A Liouville theorem in the Heisenberg group

TL;DR

This paper classifies positive solutions to a critical elliptic equation in the Heisenberg group, showing they are specific known solutions under certain conditions, extending classical results to a sub-Riemannian setting.

Contribution

It extends the classification of solutions to critical elliptic equations to the Heisenberg group, providing a Liouville theorem analogous to the Caffarelli-Gidas-Spruck result.

Findings

Solutions are Jerison-Lee's bubbles under given conditions

The classification holds for n=1 and n≥2 with control at infinity

The methods adapt classical identities and estimates to the Heisenberg group

Abstract

In this paper we classify positive solutions to the critical semilinear elliptic equation in $\mathbb{H}^n$. We prove that they are the Jerison-Lee's bubbles, provided $n=1$ or $n\geq 2$ and a suitable control at infinity holds. The proofs are based on a classical Jerison-Lee's differential identity and on pointwise/integral estimates recently obtained for critical semilinear and quasilinear elliptic equations in $\mathbb{R}^n$. In particular, the result in $\mathbb{H}^1$ can be seen as the analogue of the celebrated Caffarelli-Gidas-Spruck classification theorem.