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: ϕ u f (u)l(|∇u|)continuous functions satisfying certain monotonicity properties.The operator ϕ , 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 u f (u) in R m . We show sharpness of our conditions when we specialize to the p-Laplacian. While proving these results we obtain a strong maximum principle for ϕ 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.

No abstract

This article deals with the study of some properties of immersed curves in the conformal sphere Q n , viewed as a homogeneous space under the action of the Möbius group. After an overview on general well-known facts, we briefly focus on the links between Euclidean and conformal curvatures, in the spirit of F. Klein's Erlangen program. The core of this article is the study of conformal geodesics, defined as the critical points of the conformal arclength functional. After writing down their Euler-Lagrange equations for any n, we prove an interesting codimension reduction, namely that every conformal geodesic in Q n lies, in fact, in a totally umbilical 4-sphere Q 4 . We then extend and complete the work in Musso (Math Nachr 165:107-131, 1994) by solving the Euler-Lagrange equations for the curvatures and by providing an explicit expression even for those conformal geodesics not included in any conformal 3-sphere.

This paper deals with the study of differential inequalities with gradient terms on Carnot groups. We are mainly focused on inequalities of the form ∆ϕu ≥ f (u)l(|∇0u|), where f , l and ϕ are continuous functions satisfying suitable monotonicity assumptions and ∆ϕ is the ϕ-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 [9] 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 ∆ϕu ≥ f (u)l(|∇0u|). Finally, we show sharpness of our results for a general ϕ-Laplacian.

The aim of this paper is to study a new equivalent form of the weak maximum principle for a large class of differential operators on Riemannian manifolds. This new form has been inspired by the work of Berestycki, Hamel and Rossi, [5], for trace operators and allows us to shed new light on it and to introduce a new sufficient bounded Khas'minskii type condition for its validity. We show its effectiveness by applying it to obtain some uniqueness results in a geometric setting.