A simple proof of the optimal power in Liouville theorems

• Villegas Barranco, Salvador ^[1]
  1. [1] Universidad de Granada

     Universidad de Granada

     Granada, España

     
• Localización: Publicacions matematiques, ISSN 0214-1493, Vol. 66, Nº 2, 2022, págs. 883-892
• Idioma: inglés
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• Resumen

  • Consider the equation div(ϕ2∇σ) = 0 in RN , where ϕ > 0. It is well known [4, 2] that if there exists C > 0 such that R BR (ϕσ) 2 dx ≤ CR2 for every R ≥ 1, then σ is necessarily constant. In this paper we present a simple proof that this result is not true if we replace R2 with Rk for k > 2 in any dimension N. This question is related to a conjecture by De Giorgi.