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 Delta(phi)u >= f(u)l(vertical bar del(0)u vertical bar), where f, l and phi are continuous functions satisfying suitable monotonicity assumptions and Delta(phi) is the phi-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 Delta(phi)u >= f(u)l(vertical bar del(0)u vertical bar). Finally, we show sharpness of our results for a general phi-Laplacian. (C) 2011 Elsevier Ltd. All rights reserved.