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Liouville Type Theorems for Elliptic Equations with Gradient Terms


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.

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Correspondence to Alexander Quaas.

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S. A. was partially supported by USM Grant No. 121002 and Fondecyt grant 11110482. J. G-M and A. Q. were partially supported by Ministerio de Ciencia e Innovaci´on under grant MTM2011-27998 (Spain) and A. Q. was partially supported by Fondecyt Grant No. 1110210 and CAPDE, Anillo ACT-125. All three authors were partially supported by Programa Basal CMM, U. de Chile.

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Alarcón, S., García-Melián, J. & Quaas, A. Liouville Type Theorems for Elliptic Equations with Gradient Terms. Milan J. Math. 81, 171–185 (2013).

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Mathematics Subject Classification (2010)

  • Primary 35J60
  • Secondary 35B53


  • Liouville theorems
  • Hadamard property
  • supersolutions
  • gradient term