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 Δφu≥f(u)l(|∇0u|), where f, l and φ are continuous functions satisfying suitable monotonicity assumptions and Δφ 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 Δφu≥f(u)l(|∇0u|). Finally, we show sharpness of our results for a general φ-Laplacian.