Liouville type results and a maximum principle for non-linear differential operators on the Heisenberg group

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…”

“…We are now in a position to extend and to generalize in several directions the core of Corollary 1.4 of [10], without requiring any monotonicity on ℓ. ϕ(s) = s p−1 , p > 1, but also in the generalized mean curvature case, ϕ(s) = s(1 + s 2 ) (p−2)/2 , p ∈ (1, 2). For other elliptic operators we refer to [25,2]. The next result concerns uniqueness of radial stationary solutions of (1.1), as in Theorem 1.1 we do not require any monotonicity assumption on ℓ in R + 0 .…”

“…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…”