Qualitative properties and classification of nonnegative solutions to −Δu=f(u) in unbounded domains when f(0)<0

• Alberto Farina ^[1] ; Berardino Sciunzi ^[2]
  1. [1] University of Picardie Jules Verne

     University of Picardie Jules Verne

     Arrondissement d’Amiens, Francia

     
  2. [2] University of Calabria

     University of Calabria

     Cosenza, Italia

     
• Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 32, Nº 4, 2016, págs. 1311-1330
• Idioma: inglés
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• Resumen

  • We consider nonnegative solutions to −Δu=f(u) in unbounded Euclidean domains under zero Dirichlet boundary conditions, where f is merely locally Lipschitz continuous and satisfies f(0)<0. In the half-plane, and without any other assumption on u, we prove that u is either one-dimensional and periodic or positive and strictly monotone increasing in the direction orthogonal to the boundary. Analogous results are obtained if the domain is a strip. As a consequence of our main results, we answer affirmatively to a conjecture and to an open question posed by Berestycki, Caffarelli and Nirenberg. We also obtain some symmetry and monotonicity results in the higher-dimensional case.