On the Liouville property for fully nonlinear equations

Abstract

In this paper we prove some Liouville theorems for nonnegative viscosity supersolutions of a class of fully nonlinear uniformly elliptic problems in R N .RÉSUMÉ. -Dans ce travail nous démontrons des théorèmes de Liouville pour des sur-solutions de viscosité positives de problèmes uniformement elliptique complètement non linéaires dans R N .

Liouville results for fully nonlinear equations modeled on Hörmander vector fields: I. The Heisenberg group

“…, This extends to a different subelliptic structure some results of [18] and the more general ones in [2] for the uniformly elliptic case, as well as those in [19] for the Heisenberg group. In particular, by taking λ = Λ, our fundamental solutions of the corresponding sub-Laplacian agrees with those found by Kaplan [26].…”

“…Standard calculations similar to those carried out in [18], [19,Lemma 3.3] lead to the following Lemma 4.7. The radial functions Φ 1 (x) = ϕ 1 (ρ) and Φ 2 (x) = ϕ 2 (ρ) with…”

“…Remark 3.8. Under condition (18) the Liouville comparison principle Corollary 2.3 applies to (16) with w = log ρ and to the equation of Remark 3.8 with W = − log ρ.…”

Existence and nonexistence of subsolutions for augmented Hessian equations

“…For the Keller-Osserman condition on k−Yamabe type equations, we can see [1]. Analogous results for fully nonlinear equations have been obtained by Dolcetta, Leoni and Vitolo [4,5], Cutrì and Leoni [10], Felmer and Quaas [12], Lu and Zhu [24] and the references therein. Let the positive constant α 0 satisfy…”

Classification of supersolutions and Liouville theorems for some nonlinear elliptic problems

“…Numerous works have dealt with the question of nonexistence of supersolutions with some more general nonlinearities and operators. Without being exhaustive with the references, we mention [5], [6], [7], [8], [9], [11], [12], [15], [18], [19], [20], [28], [30] and [33] (and references therein). We also refer to the survey [29] for a more complete list.…”