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.
|Titolo:||Keller-Osserman type conditions for differential inequalities with gradient terms on the Heisenberg group|
|Parole Chiave:||Differential inequalities; Gradient term; Heisenberg group; Keller-Osserman|
|Settore Scientifico Disciplinare:||Settore MAT/05 - Analisi Matematica|
|Data di pubblicazione:||2011|
|Digital Object Identifier (DOI):||10.1016/j.jde.2011.01.006|
|Appare nelle tipologie:||01 - Articolo su periodico|