Some new Liouville-type results for fully nonlinear PDEs on the Heisenberg group

Abstract

We prove new (sharp) Liouville-type properties via degenerate Hadamard threesphere theorems for fully nonlinear equations structured over Heisenberg vector fields. As model examples, we cover the case of Pucci's extremal operators perturbed by suitable semilinear and gradient terms, extending to the Heisenberg setting known contributions valid in the Euclidean framework. Contents 1. Introduction 1 2. Preliminary notions 3 3. Maximum principles for fully nonlinear PDEs on the Heisenberg group 5 4. PDEs perturbed by semilinear terms 6 4.1. Hadamard theorems and some consequences 6 4.2. Liouville theorems 8 5. PDEs perturbed by gradient terms 13 5.1. Hadamard-type results 13 5.2. Liouville properties 14 5.3. Remarks on the assumption (22) 15 References 16