Resumen de Optimal Liouville theorems for supersolutions of elliptic equations with the Laplacian

Salomón Alarcón, Jorge José García Melián , Alexander Quaas

• In this paper we consider the question of nonexistence of positive supersolutions of the equation −1u = f (u) in exterior domains of RN , where f is continuous and positive in (0,+1). When N # 3, we find that positive supersolutions exist if and only if Z " 0 f (t) t 2(N−1) N−2 dt < +1 for some " > 0. A similar condition is found for N = 2: positive supersolutions exist if and only if Z 1 M eat f (t)dt < +1 for some a, M > 0. The proofs are extended to consider some more general operators, which include the Laplacian with gradient terms, the p-Laplacian or uniformly elliptic fully nonlinear operators with radial symmetry, like the Pucci’s extremal operatorsM±#,3, with 3 > # > 0.