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

Abstract

We study the qualitative behavior of non-negative entire solutions of differential inequalities with gradient terms on the Heisenberg group. We focus on two classes of inequalities: Δφu⩾f(u)l(|∇u|) and Δφu⩾f(u)−h(u)g(|∇u|), where f, l, h, g are non-negative continuous functions satisfying certain monotonicity properties. The operator Δφ, called the φ-Laplacian, generalizes the p-Laplace operator 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⩾f(u) in Rm. We show sharpness of our conditions when we specialize to the p-Laplacian. While proving these results we obtain a strong maximum principle for Δφ which, to the best of our knowledge, seems to be new. Our results continue to hold, with the obvious minor modifications, also for Euclidean space.