A non-smooth three critical points theorem with applications in differential inclusions

  • Alexandru Kristály
  • Waclaw Marzantowicz
  • Csaba Varga


We extend a recent result of Ricceri concerning the existence of three critical points of certain non-smooth functionals. Two applications are given, both in the theory of differential inclusions; the first one concerns a non-homogeneous Neumann boundary value problem, the second one treats a quasilinear elliptic inclusion problem in the whole \({\mathbb R^N}\).


Locally Lipschitz functions Critical points Differential inclusions 


  1. 1.
    Arcoya D., Carmona J.: A nondifferentiable extension of a theorem of Pucci-Serrin and applications. J. Differ. Equ. 235(2), 683–700 (2007)CrossRefGoogle Scholar
  2. 2.
    Bonanno G.: Some remarks on a three critical points theorem. Nonlinear Anal. 54, 651–665 (2003)CrossRefGoogle Scholar
  3. 3.
    Bonanno G.: A critical points theorem and nonlinear differential problems. J. Global Optim. 28(3–4), 249–258 (2004)CrossRefGoogle Scholar
  4. 4.
    Bonanno G., Candito P.: Non-differentiable functionals and applications to elliptic problems with discontinuous nonlinearity. J. Differ. Equ. 244(12), 3031–3059 (2008)CrossRefGoogle Scholar
  5. 5.
    Brézis H.: Analyse Fonctionnelle-Théorie et Applications. Masson, Paris (1992)Google Scholar
  6. 6.
    Chang K.C.: Variational methods for non-differentiable functionals and their applications to partial differential equations .J. Math. Anal. Appl. 80, 102–129 (1981)CrossRefGoogle Scholar
  7. 7.
    Clarke F.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)Google Scholar
  8. 8.
    Krawcewicz W., Marzantowicz W.: Some remarks on the Lusternik-Schnirelman method for non- differentiable functionals invariant with respect to a finite group action. Rocky Mt. J. Math. 20, 1041–1049 (1990)CrossRefGoogle Scholar
  9. 9.
    Kristály A.: Infinitely many solutions for a differential inclusion problem in \({\mathbb{R}^N}\) . J. Differ. Equ. 220, 511–530 (2006)CrossRefGoogle Scholar
  10. 10.
    Marano S.A., Motreanu D.: On a three critical points theorem for non-differentiable functions and applications to nonlinear boundary value problems. Nonlinear Anal. 48, 37–52 (2002)CrossRefGoogle Scholar
  11. 11.
    Marano S.A., Papageorgiou N.S.: On a Neumann problem with p-Laplacian and non-smooth potential. Differ. Integral Equ. 19(11), 1301–1320 (2006)Google Scholar
  12. 12.
    Motreanu D., Varga Cs.: Some critical point results for locally Lipschitz functionals. Comm. Appl. Nonlinear Anal. 4, 17–33 (1997)Google Scholar
  13. 13.
    Pucci P., Serrin J.: A mountain pass theorem. J. Differ. Equ. 60, 142–149 (1985)CrossRefGoogle Scholar
  14. 14.
    Ricceri B.: On a three critical points theorem. Arch. Math. (Basel) 75, 220–226 (2000)Google Scholar
  15. 15.
    Ricceri B.: Existence of three solutions for a class of elliptic eigenvalue problems. Math. Comput. Model. 32, 1485–1494 (2000)CrossRefGoogle Scholar
  16. 16.
    Ricceri B.: Three solutions for a Neumann problem. Topol. Methods Nonlinear Anal. 20, 275–282 (2002)Google Scholar
  17. 17.
    Ricceri, B.: A three critical points theorem revisited. Nonlinear Anal. doi: 10.1016/
  18. 18.
    Ricceri B.: Minimax theorems for limits of parametrized functions having at most one local minimum lying in a certain set. Topol. Appl. 153, 3308–3312 (2006)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  • Alexandru Kristály
    • 1
  • Waclaw Marzantowicz
    • 2
  • Csaba Varga
    • 3
  1. 1.Department of EconomicsBabeş-Bolyai UniversityCluj-NapocaRomania
  2. 2.Department of MathematicsAdam Mickiewicz UniversityPoznanPoland
  3. 3.Department of Mathematics and Computer ScienceBabeş-Bolyai UniversityCluj-NapocaRomania

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