Liouville type results and a maximum principle for non-linear differential operators on the Heisenberg group

We prove Liouville type results for non-negative solutions of the differential inequality Δ φ u ⩾ f ( u ) ℓ ( | ∇ 0 u | ) on the Heisenberg group under a generalized Keller–Osserman condition. The operator Δ φ u is the φ-Laplacian defined by div 0 ( | ∇ 0 u | − 1 φ ( | ∇ 0 u | ) ∇ 0 u ) and φ, f and ℓ satisfy mild structural conditions. In particular, ℓ is allowed to vanish at the origin. A key tool that can be of independent interest is a strong maximum principle for solutions of such differential inequality.