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 $Δ^ϕu \ge f(u)l(|\nabla u|)$ and $Δ^ϕu \ge f(u) - h(u) g(|\nabla u|)$, where $f,l,h,g$ are non-negative continuous functions satisfying certain monotonicity properties. The operator $Δ^ϕ$, called the $ϕ$-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 $Δu \ge f(u)$ 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.

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