On the Classification of Stable Solutions of the Fractional Equation

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Abstract

In this paper, we study the nonexistence result for the following nonlinear elliptic equation:
$$(-{\Delta})^{s} u+\lambda u= |u|^{p-1}u \; \;\text{ in} \; \mathbb{R}^{n} $$
where n ≥ 2s, 0 < s < 1, λ > 0 and p > 1. 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

Morse index Liouville-type theorems Pohozaev identity Monotonicity formula 

Mathematics Subject Classification 2010

Primary: 35J55, 35J65 Secondary: 35B65 

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Author Contributions

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

Compliance with Ethical Standards

Competing of interests

The authors declare that they have no competing interests.

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Faculté des Sciences, Département de MathématiquesUniversité de SfaxSfaxTunisia

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