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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.

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Authors and Affiliations

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

    Belgacem Rahal & Cherif Zaidi

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  1. Belgacem Rahal
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  2. Cherif Zaidi
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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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Correspondence to Belgacem Rahal.

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Rahal, B., Zaidi, C. On the Classification of Stable Solutions of the Fractional Equation. Potential Anal 50, 565–579 (2019). https://doi.org/10.1007/s11118-018-9694-6

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  • DOI: https://doi.org/10.1007/s11118-018-9694-6

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Mathematics Subject Classification 2010

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