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Separable infinity harmonic functions in cones

  • Marie-Françoise Bidaut-Véron
  • Marta Garcia-Huidobro
  • Laurent Véron
Article
  • 64 Downloads

Abstract

We study the existence of separable infinity harmonic functions in any cone of \(\mathbb R^N\) vanishing on its boundary under the form \(u(r,\sigma )=r^{-\beta }\psi (\sigma )\). We prove that such solutions exist, the spherical part \(\psi \) satisfies a nonlinear eigenvalue problem on a subdomain of the sphere \(S^{N-1}\) and that the exponents \(\beta =\beta _+>0\) and \(\beta =\beta _-<0\) are uniquely determined if the domain is smooth. We extend some of our results to non-smooth domains.

Mathematics Subject Classification

35D40 35J70 35J62 

Notes

Acknowledgements

This article has been prepared with the support of the collaboration programs ECOS C14E08 and FONDECYT grant 1160540 for the three authors. The authors are grateful to the referee for a careful reading of their work.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Marie-Françoise Bidaut-Véron
    • 1
  • Marta Garcia-Huidobro
    • 2
  • Laurent Véron
    • 1
  1. 1.Laboratoire de Mathématiques et Physique ThéoriqueUniversité Francois RabelaisToursFrance
  2. 2.Departamento de MathematicàPontificia Università CatolicaSantiagoChile

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