A Liouville theorem in the Heisenberg group

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} .