Set-Valued and Variational Analysis

, Volume 19, Issue 2, pp 311–327 | Cite as

Three Solutions for a Partial Differential Inclusion Via Nonsmooth Critical Point Theory

  • Antonio IannizzottoEmail author


The existence of three solutions for a partial differential inclusion involving a perturbed nonlinearity and two real parameters is proved. Moreover, an estimate of the norms of solutions, independent of both the parameters and the perturbation, is achieved. The main theoretical tool is an extension to nonsmooth functionals of a very recent three critical points theorem of Ricceri.


Partial differential inclusions Nonsmooth critical point theory 

Mathematics Subject Classifications (2010)

34A60 49J52 58E05 


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  1. 1.
    Anello, G.: A multiplicity theorem for critical points of functionals on reflexive Banach spaces. Arch. Math. (Basel) 82, 172–179 (2004)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Chang, K.C.: Variational methods for nondifferentiable functionals and their applications to partial differential equations. J. Math. Anal. Appl. 80, 102–129 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)zbMATHGoogle Scholar
  4. 4.
    Degiovanni, M., Marzocchi, M.: A critical point theory for nonsmooth functionals. Ann. Math. Pures Appl. 167, 73–100 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Faraci, F., Iannizzotto, A.: An extension of a multiplicity theorem by Ricceri with an application to a class of quasilinear equations. Stud. Math. 172, 275–287 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Frigon, M.: On a critical point theory for multivalued functionals and application to partial differential inclusions. Nonlinear Anal. 31, 735–753 (1998)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Gasiński, L., Papageorgiou, N.S.: Nonlinear Analysis. Chapman & Hall/CRC, Boca Raton (2006)zbMATHGoogle Scholar
  8. 8.
    Iannizzotto, A.: Three critical points for perturbed nonsmooth functionals and applications. Nonlinear Anal. 72, 1319–1338 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Ribarska, N., Tsachev, T., Krastanov, M.: A note on: “On a critical point theory for multivalued functionals and application to partial differential inclusions”. Nonlinear Anal. 43, 153–158 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Kristály, A., Marzantowicz, W., Varga, Cs.: A non–smooth three critical points theorem with applications in differential inclusions. J. Glob. Optim. 46, 49–62 (2010)zbMATHCrossRefGoogle Scholar
  11. 11.
    Motreanu, D., Panagiotopoulos, P.D.: Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities. Kluwer, Dordrecht (1999)zbMATHGoogle Scholar
  12. 12.
    Naselli, O.: A class of functionals on a Banach space for which strong and weak local minima do coincide. Optimization 50, 407–411 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Ricceri, B.: Sublevel sets and global minima of coercive functionals and local minima of their perturbations. J. Nonlinear Convex Anal. 5, 157–168 (2004)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Ricceri, B.: A further three critical points theorem. Nonlinear Anal. 71, 4151–4157 (2009)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Dipartimento di Matematica e InformaticaUniversità degli Studi di CataniaCataniaItaly

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