Optimal Liouville theorems for supersolutions of elliptic equations with the Laplacian

• Salomón Alarcón ^[1] ; Jorge García-Melián ^[2] ; Alexander Quaas ^[1]
  1. [1] Universidad Técnica Federico Santa María

     Universidad Técnica Federico Santa María

     Valparaíso, Chile

     
  2. [2] Universidad de La Laguna

     Universidad de La Laguna

     San Cristóbal de La Laguna, España

     
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 16, Nº 1, 2016, págs. 129-158
• Idioma: inglés
• DOI: 10.2422/2036-2145.201402_007
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• Resumen

  • 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.