Calculus of Variations and Geometric Measure Theory

G. Catino - Y. Li - D. D. Monticelli - A. Roncoroni

A Liouville theorem in the Heisenberg Group

created by catino on 17 Oct 2023


Submitted Paper

Inserted: 17 oct 2023
Last Updated: 17 oct 2023

Year: 2023


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.