A note on Keller-Osserman conditions on Carnot groups

This paper deals with the study of differential inequalities with gradient terms on Carnot groups. We are mainly focused on inequalities of the form Δ_φu≥f(u)l(|∇₀u|), where f, l and φ are continuous functions satisfying suitable monotonicity assumptions and Δ_φ is the φ-Laplace operator, a natural generalization of the p-Laplace operator which has recently been studied in the context of Carnot groups. We extend to general Carnot groups the results proved in Magliaro et al. (2011) [7] for the Heisenberg group, showing the validity of Liouville-type theorems under a suitable Keller-Osserman condition. In doing so, we also prove a maximum principle for inequality Δ_φu≥f(u)l(|∇₀u|). Finally, we show sharpness of our results for a general φ-Laplacian.