Work

A Liouville theorem in the Heisenberg group

Year: 2025

Type: article

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} .

Source: Journal of the European Mathematical Society

DOI: 10.4171/jems/1705

Authors: Giovanni Catino, Yanyan Li, Dario D. Monticelli, Alberto Roncoroni

Institutions: Politecnico di Milano, Rutgers, The State University of New Jersey

Language: English

Topic: Nonlinear Partial Differential Equations

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: National Science Foundation, Istituto Nazionale di Alta Matematica "Francesco Severi", Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni, Gruppo Nazionale per le Strutture Algebriche, Geometriche e le loro Applicazioni

Awards: G2528524749, G5245466801, Nonlinear Elliptic Equations and Systems, and Applications