A Liouville theorem in the Heisenberg group

Abstract

In this paper, we classify positive solutions to the critical semilinear elliptic equation in the Heisenberg group ${\mathbb{H}}^1$. We prove that they are the Jerison–Lee’s bubbles. We do not require any finite energy or a priori symmetry assumptions. The proof is based on a classical Jerison–Lee’s differential identity and on new pointwise/integral estimates, in the spirit of recent developments for critical semilinear and quasilinear elliptic equations in ${\mathbb{R}}^n$. In particular, our result is the analogue for ${\mathbb{H}}^1$ of the celebrated Caffarelli–Gidas–Spruck classification theorem in ${\mathbb{R}}^n$.