Milan Journal of Mathematics

, Volume 81, Issue 1, pp 171–185 | Cite as

Liouville Type Theorems for Elliptic Equations with Gradient Terms

  • Salomón Alarcón
  • Jorge García-Melián
  • Alexander QuaasEmail author


In this paper we obtain Liouville type theorems for nonnegative supersolutions of the elliptic problem \({-\Delta u + b(x)|\nabla u| = c(x)u}\) in exterior domains of \({\mathbb{R}^N}\). We show that if lim \({{\rm inf}_{x \longrightarrow \infty} 4c(x) - b(x)^2 > 0}\) then no positive supersolutions can exist, provided the coefficients b and c verify a further restriction related to the fundamental solutions of the homogeneous problem. The weights b and c are allowed to be unbounded. As an application, we also consider supersolutions to the problems \({-\Delta u + b|x|^{\lambda}|{\nabla} u| = c|x|^{\mu} u^p}\) and \({-\Delta u + be^{\lambda |x|}|\nabla u| = ce^{\mu |x|}u^p}\), where p > 0 and λ, μ ≥ 0, and obtain nonexistence results which are shown to be optimal.

Mathematics Subject Classification (2010)

Primary 35J60 Secondary 35B53 


Liouville theorems Hadamard property supersolutions gradient term 


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Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Salomón Alarcón
    • 1
  • Jorge García-Melián
    • 2
    • 3
  • Alexander Quaas
    • 4
    Email author
  1. 1.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaísoChile
  2. 2.Departamento de Análisis MatemáticoUniversidad de La LagunaLa LagunaSpain
  3. 3.Instituto Universitario de Estudios Avanzados (IUdEA) en Física Atómica, Molecular y Fotónica, Facultad de FísicaUniversidad de La LagunaLa LagunaSpain
  4. 4.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaísoChile

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