A Liouville theorem in the Heisenberg group

2023/10/16 by Catino, Giovanni, Li, Yanyan, Monticelli, Dario D. +1 · 4 citations
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2310.10469

Abstract

In this paper we classify positive solutions to the critical semilinear elliptic equation in ℍⁿ. We prove that they are the Jerison-Lee's bubbles, provided n=1 or n≥ 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 ℝⁿ. In particular, the result in ℍ¹ can be seen as the analogue of the celebrated Caffarelli-Gidas-Spruck classification theorem.

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• CR Yamabe Equation on the Heisenberg Group via the method of moving spheres
• A Liouville theorem for the CR Yamabe type equation on Sasakian manifolds
• A Liouville theorem for the 2-Hessian equation on the Heisenberg group
• Liouville type results for semilinear elliptic equation and inequality on pseudo-Hermitian manifolds

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