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 Δφu⩾f(u)ℓ(|∇0u|) on the Heisenberg group under a generalized Keller-Osserman condition. The operator Δφu is the φ-Laplacian defined by div0(|∇0u|−1φ(|∇0u|)∇0u) 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.