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On the Classification of Stable Solutions of the Fractional Equation

  • Belgacem Rahal
  • Cherif Zaidi
Article

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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Notes

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.

References

  1. 1.
    Caffarelli, L., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Comm. Pure Appl. Math. 42(3), 271–297 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Comm. Partial Differ. Equ. 32(7-9), 1245–1260 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chen, W., Li, C., Ou, B.: Classification of solutions for an integral equation. Comm. Pure Appl. Math. 59(3), 330–343 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Davila, J., Dupaigne, L., Wei, J.: On the fractional Lane-Emden equation. Trans. Amer. Math. Soc. 369(9), 6087–6104 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Fabes, E.B., Kenig, C.E., Serapioni, R.P.: The local regularity of solutions of degenerate elliptic equations. Comm Partial Differ. Equ. 7(1), 77–116 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Farina, A.: On the classification of solutions of the Lane-Emden equation on unbounded domains of \(\mathbb {R}^{n}\). J. Math. Pures Appl. (9) 87(5), 537–561 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gidas, B., Spruck, J.: A priori bounds for positive solutions of nonlinear ellip- tic equations. Comm. Partial Differ. Equ. 6(8), 883–901 (1981)CrossRefzbMATHGoogle Scholar
  8. 8.
    Harrabi, A., Rahal, B.: On the sixth-order JosephLundgren exponent. J. Annales Henri Poincaré 18(3), 1055–1094 (2017)CrossRefzbMATHGoogle Scholar
  9. 9.
    Harrabi, A., Rahal, B.: Liouville type theorems for elliptic equations in half-space with mixed boundary value conditions, J. Advances in Nonlinear Analysis.  https://doi.org/10.1515/anona-2016-0168
  10. 10.
    Harrabi, A., Rahal, B.: Liouville results for m-Laplace equations in half-space and strips with mixed boundary value conditions and finite Morse index, J. Dyn. Diff. Equat.  https://doi.org/10.1007/s10884-017-9593-3
  11. 11.
    Harrabi, A., Selmi, A., Zaidi, C.: A Liouville type-theorems for an elliptic equation, preprintGoogle Scholar
  12. 12.
    Joseph, D.D., Lundgren, T.S.: Quasilinear Dirichlet problems driven by positive sources. Arch. Rational Mech. Anal. 49, 241–269 (1972/73)Google Scholar
  13. 13.
    Molcanov, S.A., Ostrovskii, E.: Symmetric stable processes as traces of degenerate diffusion processes. Teor Verojatnost. i Primenen. 14, 127–130 (1969)MathSciNetGoogle Scholar
  14. 14.
    Ros-Oton, X., Serra, J.: The Pohozaev identity for the fractional Laplacian. Arch. Ration. Mech. Anal. 213(2), 587–628 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Spitzer, F.: Some theorems concerning 2-dimensional Brownian motion. Trans. Amer. Math. Soc. 87, 187–197 (1958)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Li, Y.: Remark on some conformally invariant integral equations: the method of moving spheres. J. Eur. Math. Soc. (JEMS) 6(2), 153–180 (2004)MathSciNetCrossRefzbMATHGoogle Scholar

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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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