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: 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.

The compact support principle for differential inequalities with gradient terms

“…Note that (KF) is the ''dual'' of the Keller-Osserman condition studied in [9][10][11]. It is well known, see for instance Theorem 1.1 in [5], that (KF) is necessary for the validity of the Compact Support Principle for the differential inequality Lv ≥ f (v) .…”

Nonlinear elliptic inequalities with gradient terms on the Heisenberg group

“…For further generalization to quasilinear inequalities, possibly with singular of degenerate weights, we refer to [7][8][9][10]21,22]. The first result in this direction, but in the Heisenberg group setting, can be found in [17,2]. Recently, this has been extended to the Carnot groups in [1], adding further restrictions due to the presence of a new term which arises since the norm is not ∞-harmonic in that setting.…”

“…Since we are interested in nonnegative entire solutions of elliptic coercive inequalities in all the space, as in [10,17,2] we make use of an appropriate generalized Keller-Osserman condition for inequality (1.2). To this aim we also assume throughout the paper that…”

“…1. In what follows we assume monotonicity on f . In particular in the next theorem we require strict monotonicity on f , similarly as in [8,17,1,2]. Indeed, this assumption is due to the technique used, that is to an argument involving a comparison theorem.…”

“…Recently, in [17,2] results similar to Theorem 1.3 are given when ℓ is C -monotone nondecreasing in R + 0 , that is there exists…”