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

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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.

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Notes

  1. with a standard abuse of notation, we have used the same letter u to denote distinct functions.

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Acknowledgements

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

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Authors and Affiliations

  1. Département de Mathématiques, Faculté des Sciences, Université de Sfax, B.P 1171, 3000, Sfax, Tunisia

    Belgacem Rahal

  2. Institut Supérieur des Mathématiques Appliquées et de l’Informatique de Kairouan, Université de Kairouan, Kairouan, Tunisia

    Abdellaziz Harrabi

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  1. Belgacem Rahal
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  2. Abdellaziz Harrabi
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Correspondence to Belgacem Rahal.

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

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Rahal, B., Harrabi, A. Liouville Results for m-Laplace Equations in Half-Space and Strips with Mixed Boundary Value Conditions and Finite Morse Index. J Dyn Diff Equat 30, 1161–1185 (2018). https://doi.org/10.1007/s10884-017-9593-3

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  • DOI: https://doi.org/10.1007/s10884-017-9593-3

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