Nonexistence results for elliptic differential inequalities with a potential on Riemannian manifolds

Abstract

In this paper we are concerned with a class of elliptic differential inequalities with a potential both on and on Riemannian manifolds. In particular, we investigate the effect of the geometry of the underlying manifold and of the behavior of the potential at infinity on nonexistence of nonnegative solutions.

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Acknowledgments

The authors wish to thank Prof. Marco Rigoli for interesting discussions, and in particular, for suggesting Section 4, and Prof. Lorenzo D’Ambrosio and Prof. Alexander Grigor’yan for having brought to their attention the works by Sun [22] and [23]. Moreover, the authors also wish to thank the anonymous referee for his helpful suggestions and comments.

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Correspondence to F. Punzo.

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The three authors are supported by GNAMPA project “Analisi globale ed operatori degeneri” and are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

Communicated by A. Malchiodi.

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Mastrolia, P., Monticelli, D.D. & Punzo, F. Nonexistence results for elliptic differential inequalities with a potential on Riemannian manifolds. Calc. Var. 54, 1345–1372 (2015). https://doi.org/10.1007/s00526-015-0827-0

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Mathematics Subject Classification

  • 53C20
  • 53C25
  • 53A55