Keller-Osserman type conditions for differential inequalities with gradient terms on the Heisenberg group

The aim of this paper is to study the qualitative behaviour of non-negative entire solutions of certain differential inequalities involving gradient terms on the Heisenberg group. We focus our investigation on the two classes of inequalities of the form ${\mathrm{\Delta }}^{\varphi }u\ge f(u)l(\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid })$ and $\Delta^\phi u \ge f(u) - h(u) g(|\nabla u|)$,where$f,l,h,g$ are non-negative continuous functions satisfying certain monotonicity properties. The operator ${\mathrm{\Delta }}^{\varphi }$, called the $\varphi$-Laplacian, can be viewed as a natural generalization of the $p$-Laplace operator recently considered by various authors in this setting. We prove some Liouville theorems introducing two new Keller-Osserman type conditions, both extending the classical one which appeared long ago in the study of the prototype differential inequality $\Delta u \ge f(u)$in$\erre^m$. Furthermore, we show sharpness of our conditions when we specialize to the case of the $p$-Laplacian. Needless to say, our results continue to hold, with the obvious minor modifications, also in the Euclidean space.