Fractional eigenvalues

  • Erik Lindgren
  • Peter Lindqvist


We study the non-local eigenvalue problem
$$\begin{aligned} 2\, \int \limits _{\mathbb{R }^n}\frac{|u(y)-u(x)|^{p-2}\bigl (u(y)-u(x)\bigr )}{|y-x|^{\alpha p}}\,dy +\lambda |u(x)|^{p-2}u(x)=0 \end{aligned}$$
for large values of \(p\) and derive the limit equation as \(p\rightarrow \infty \). Its viscosity solutions have many interesting properties and the eigenvalues exhibit a strange behaviour.

Mathematics Subject Classification (2000)

35J60 35P30 35R11 



We thank Evgenia Malinnikova for helping us to verify an inequality. We thank the referees for a careful reading of the manuscript and for drawing our attention to the article [9].


  1. 1.
    Anane, A.: Simplicité et isolation de la première valeur propre du p-laplacien avec poids. C. R. Acad. Sci. Paris Sér. I Math. 305(16), 725–728 (1987)Google Scholar
  2. 2.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations, Systems & Control: Foundations & Applications. Birkhäuser Boston Inc., Boston, MA (1997). With appendices by Maurizio Falcone and Pierpaolo SoraviaGoogle Scholar
  3. 3.
    Belloni, M., Kawohl, B.: A direct uniqueness proof for equations involving the p-Laplace operator. Manuscripta Math. 109(2), 229–231 (2002)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Bourgain, J., Brezis, H., Mironescu, P.: Limiting embedding theorems for \({\rm W^{s, p}}\) when \(s\uparrow 1\) and applications. J. Anal. Math. 87, 77–101. Dedicated to the memory of Thomas H. Wolff (2002)Google Scholar
  5. 5.
    Chambolle, A., Lindgren, E., Monneau, R.: A Hölder infinity Laplacian, accepted for publication in ESAIM: Control, Optimisation and Calculus of Variations (2011)Google Scholar
  6. 6.
    Champion, T., De Pascale, L., Jimenez, C.: The \(\infty \)-eigenvalue problem and a problem of optimal transportation. Commun. Appl. Anal. 13(4), 547–565 (2009)Google Scholar
  7. 7.
    Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhikers guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Frank, R.L., Leander, G.: Refined semiclassical asymptotics for fractional powers of the Laplace operator. preprint (2011)Google Scholar
  9. 9.
    Fukagai, N., Ito, M., Narukawa, K.: Limit as \(p\rightarrow \infty \) of p-Laplace eigenvalue problems and L\(^\infty \)-inequality of the Poincaré type. Differ. Integr. Equ. 12(2), 183–206 (1999)MATHMathSciNetGoogle Scholar
  10. 10.
    Hynd, R., Smart, C.K., Yu, Y.: Nonuniqueness of infinity ground states, preprint (2012)Google Scholar
  11. 11.
    Ishii, H., Nakamura, G.: A class of integral equations and approximation of p-Laplace equations. Calc. Var. Partial Differ. Equ. 37(3–4), 485–522 (2010)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Juutinen, P., Lindqvist, P., Manfredi, J.J.: The \(\infty \)-eigenvalue problem. Arch. Ration. Mech. Anal. 148(2), 89–105 (1999)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Kassmann, M.: The classical Harnack inequality fails for nonlocal operators. preprint No. 360, Sonderforschungsbereich 611Google Scholar
  14. 14.
    Kawohl, B., Lindqvist, P.: Positive eigenfunctions for the p-Laplace operator revisited. Analysis (Munich) 26(4), 545–550 (2006)MATHMathSciNetGoogle Scholar
  15. 15.
    Kellogg, O.D.: Foundations of Potential Theory, Reprint from the first edition of 1929: Die Grundlehren der Mathematischen Wissenschaften, Band 31. Springer, Berlin (1967)Google Scholar
  16. 16.
    Koike, S.: A Beginner’s Guide to the Theory of Viscosity Solutions, vol. 13. MSJ Memoirs, Mathematical Society of Japan, Tokyo (2004)MATHGoogle Scholar
  17. 17.
    Kwaśnicki, M.: Eigenvalues of the fractional Laplace operator in the interval. J. Funct. Anal. 262(5), 2379–2402 (2012)CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Ôtani, M., Teshima, T.: On the first eigenvalue of some quasilinear elliptic equations. Proc. Jpn. Acad. Ser. A Math. Sci. 64(1), 8–10 (1988)Google Scholar
  19. 19.
    Yifeng, Y.: Some properties of the ground states of the infinity Laplacian. Indiana Univ. Math. J. 56(2), 947–964 (2007)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Zoia, A., Rosso, A., Kardar, M.: Fractional Laplacian in bounded domains. Phys. Rev. E (3) 76(2), 021116, 11 (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and TechnologyTrondheimNorway

Personalised recommendations