A Liouville theorem in the Heisenberg group

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 $n\ge 2$ 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.