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Journal of Dynamics and Differential Equations

, Volume 30, Issue 3, pp 1161–1185 | Cite as

Liouville Results for m-Laplace Equations in Half-Space and Strips with Mixed Boundary Value Conditions and Finite Morse Index

  • Belgacem Rahal
  • Abdellaziz Harrabi
Article

Abstract

We consider sign-changing solutions of the equation \(-\Delta _m u= |u|^{p-1}u\) where \(m\ge 2\) and \(p>1\) in half-space and strips with nonlinear mixed boundary value conditions. We prove Liouville type theorems for stable solutions or for solutions which are stable outside a compact set. The main methods used are the integral estimates, the Pohozaev-type identity and the monotonicity formula.

Keywords

Liouville theorem Stable solutions Mixed boundary value conditions Stability outside a compact set Pohozaev identity Monotonicity formula 

Notes

Acknowledgements

The authors would like to thank Professor Louis Dupaigne for reading carefully the manuscript and for many helpful comments.

Compliance with Ethical Standards

Conflict of interest

The authors declare that they have no conflict of interest.

Author’s Contributions

All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Département de Mathématiques, Faculté des SciencesUniversité de SfaxSfaxTunisia
  2. 2.Institut Supérieur des Mathématiques Appliquées et de l’Informatique de KairouanUniversité de KairouanKairouanTunisia

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