Elliptic equations with diffusion parameterized by the range of nonlocal interactions



We consider quasilinear elliptic equations where the diffusion at each point depends on all the values of the solution in a neighborhood of this point. The size of this neighborhood is parameterized by some non-negative number which represents the range of nonlocal interactions. For fixed values of the parameter, the issue of the existence and local uniqueness of the solution is addressed. In a radial symmetric setting, we give pointwise estimates of the solutions and prove the existence of multiple solutions. Regarding bifurcation theory, we show that many local branches of solutions may exist while, among them, only one is global and has no bifurcation point.

Mathematics Subject Classification (2000)

35J60 35B30 


Nonlocal elliptic equations Branches of solutions Parameter Nonlocal diffusion 


  1. 1.
    Ambrosetti A., Giovanni P.: A Primer of Nonlinear Analysis, vol. 34 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (1993)Google Scholar
  2. 2.
    Chasseigne E., Chaves M., Rossi J.D.: Asymptotic behavior for nonlocal diffusion equations. J. Math. Pures Appl. (9) 86(3), 271–291 (2006)MATHMathSciNetGoogle Scholar
  3. 3.
    Chipot M., Lovat B.: On the asymptotic behaviour of some nonlocal problems. Positivity 3(1), 65–81 (1999)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chipot M., Molinet L.: Asymptotic behaviour of some nonlocal diffusion problems. Appl. Anal. 80(3-4), 279–315 (2001)MATHMathSciNetGoogle Scholar
  5. 5.
    Medeiros L.A., Limaco J., Menezes S.B.: Vibrations of elastic strings: mathematical aspects. I. J. Comput. Anal. Appl. 4(2), 91–127 (2002)MATHMathSciNetGoogle Scholar
  6. 6.
    Zhang Z., Perera K.: Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow. J. Math. Anal. Appl. 317(2), 456–463 (2006)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag 2009

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et Applications, UMR 6086Université de Poitiers & CNRS, SP2MIFuturoscope Chasseneuil CedexFrance

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