Advertisement

Potential Analysis

, Volume 45, Issue 1, pp 187–200 | Cite as

Liouville Theorems for a General Class of Nonlocal Operators

  • Mouhamed Moustapha Fall
  • Tobias WethEmail author
Article

Abstract

In this paper, we study the equation \(\mathcal {L} u=0\) in \(\mathbb {R}^{N}\), where \(\mathcal {L}\) belongs to a general class of nonlocal linear operators which may be anisotropic and nonsymmetric. We classify distributional solutions of this equation, thereby extending and generalizing recent Liouville type theorems in the case where \(\mathcal {L}= (-{\Delta })^{s}\), s ∈ (0, 1) is the classical fractional Laplacian.

Keywords

Nonlocal elliptic equations Liouville theorem Anisotropic operator 

Mathematics Subject Classification (2010)

35R11 35B40 35J75 47B25 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Albert, J.P., Bona, J.L., Saut, J.-C.: Model equations for waves in stratified fluids. Proc. R. Soc. Lond. A 453, 1233–1260 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Abatangelo, N.: Large s-harmonic functions and boundary blow-up solutions for the fractional laplacian. Discrete Contin. Dyn. Syst. A 35(12), 5555–5607 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bogdan, K., Kulczycki, T., Nowak, A.: Gradient estimates for harmonic and q-harmonic functions of symmetric stable processes. Illinois J. Math. 46(2), 541–556 (2002)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Chen, W., D’Ambrosio, L., Li, Y.: Some Liouville theorems for the fractional Laplacian. Nonlin. Anal. 121, 370–381 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    De Carli, L.: Local L p inequalities for Gegenbauer polynomials. In: Topics in Classical Analysis and Applications in Honor of Daniel Waterman, pp. 73–87. World Sci. Publ., Hackensack (2008)CrossRefGoogle Scholar
  6. 6.
    Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi, F.: Higher Transcendental Functions, vol. 2. McGraw-Hill, New York (1953)zbMATHGoogle Scholar
  7. 7.
    Eskin, G.: Lectures on Linear Partial Differential Equations Graduate Studies in Mathematics, vol. 123. American Mathematical Society, Providence, Rhode Island (2011)Google Scholar
  8. 8.
    Fall, M.M., Felli, V.: Unique continuation properties for the relativistic Shrördinger operator with singular potential. Discrete Contin. Dyn. Syst. A, V. 35 (12), 5827–5867 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Fall, M.M.: Entire s-harmonic functions are affine. Proc. Amer. Math. Soc. (In press)Google Scholar
  10. 10.
    Ferrari, F., Verbitsky, I.: Radial fractional Laplace operators and Hessian inequalities. J. Differ. Equ. 253(1), 244–272 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Frank, R.: On the uniqueness of ground states of non-local equations. J. É. D. P. Exposé no V, 10 p (2011)Google Scholar
  12. 12.
    Han, Q., Lin, F.H.: Elliptic partial differential equations. Courant Lecture Notes in Mathematics, 1. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence RI (1997)Google Scholar
  13. 13.
    Ros-Oton, X., Serra, J.: Regularity theory for general stable operators. Preprint (2014). arXiv:1412.3892
  14. 14.
    Rubin, B.: Inversion of fractional integrals related to the spherical Radon transform. J. Funct. Anal. 157(2), 470–487 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Zhuo, R., Chen, W., Cui, X., Yuan, Z.: A Liouville theorem for the fractional Laplacian. Preprint (2014). arXiv:1401.7402
  16. 16.
    Szegö, G.: Orthogonal Polynomials, vol. 23. Amer. Math. Soc., Coll. Publ., New York (1959)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.African Institute for Mathematical Sciences (A.I.M.S.) of SenegalMbourSénégal
  2. 2.Goethe-Universität Frankfurt, Institut für MathematikFrankfurtGermany

Personalised recommendations