Rendiconti del Circolo Matematico di Palermo

, Volume 55, Issue 1, pp 113–122 | Cite as

Multiplicity results for a Neumann problem withp-Laplacian and non-smooth potential



A multiplicity theorem for a non-smooth homogeneous Neumann problem withp-Laplacian is established through a locally Lipschitz continuous version of the Brézis-Nirenberg critical point result in presence of splitting. Some special cases are then pointed out.

Key words phrases

p-Laplacian elliptic hemivariational inequalities elliptic equations with discontinuous nonlinearities multiple solutions 

2000 Mathematics Subject Classification

35J20 35J85 49J40 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Brézis H.,Analyse Fonctionnelle — Théorie et Applications, Masson, Paris, 1983.MATHGoogle Scholar
  2. [2]
    Brézis H., Nirenberg L.,Remarks on finding critical points, Comm. Pure Appl. Math.,44 (1991), 939–963.CrossRefMathSciNetMATHGoogle Scholar
  3. [3]
    Candito P.,Infinitely many solutions to the Neumann problem for elliptic equations involving the p-Laplacian and with discontinuous nonlinearities, Proc. Edinburgh Math. Soc.,45 (2002), 397–409.CrossRefMathSciNetMATHGoogle Scholar
  4. [4]
    Casas E., Fernández L.,A Green’s formula for quasilinear elliptic operators, J. Math. Anal. Appl.,142 (1989), 62–73.CrossRefMathSciNetMATHGoogle Scholar
  5. [5]
    Chang K.-C.,Variational methods for nondifferentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl.,80 (1981), 102–129.CrossRefMathSciNetMATHGoogle Scholar
  6. [6]
    Clarke F. H.,Optimization and Nonsmooth Analysis, Classics Appl. Math., vol. 5, SIAM, Philadelphia, 1990.MATHGoogle Scholar
  7. [7]
    De Giorgi E., Buttazzo G., Dal Maso G.,On the lower semicontinuity of certain integral functionals, Atti Accad. Naz. Lincei C1. Sci. Fis. Mat. Natur. Rendiconti Lincei (8) Mat. Appl.,74 (1983), 274–282.MATHGoogle Scholar
  8. [8]
    Denkowski Z., Migorski S., Papageorgiou N. S.,An Introduction to Nonlinear Analysis: Theory, Kluwer/Plenum, New York, 2003.Google Scholar
  9. [9]
    Gasiński L., Rapageorgiou N. S.,Nonsmooth Critical Point Theory and Nonlinear Boundary Value Problems, Ser. Math. Anal. Appl., 8, Chapman and Hall/CRC Press, Boca Raton, 2005.MATHGoogle Scholar
  10. [10]
    Livrea R., Marano S. A., Motreanu D.,Critical points for nondifferentiable functions in presence of splitting, J. Differential Equations, to appear.Google Scholar
  11. [11]
    Marano S. A.,Elliptic boundary-value problems with discontinuous nonlinearities, Set-Valued Anal.,3 (1995), 167–180.CrossRefMathSciNetMATHGoogle Scholar
  12. [12]
    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 (2002), 37–52.CrossRefMathSciNetMATHGoogle Scholar
  13. [13]
    Marano S. A., Papageorgiou N. S.,On a Neumann problem with p-Laplacian and non-smooth potential, submitted for publication.Google Scholar
  14. [14]
    Motreanu D., Panagiotopoulos P. D.,Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, Nonconvex Optim Appl.,29, Kluwer, Dordrecht, 1998.Google Scholar
  15. [15]
    Motreanu D., Radulescu V.,Variational and Non-variational Methods in Nonlinear Analysis and Boundary Value Problems, Nonconvex Optim. Appl.,67, Kluwer, Dordrecht, 2003.MATHGoogle Scholar
  16. [16]
    Rabinowitz P. H.,Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math.,65, Amer. Mat. Soc., Providence, 1986.Google Scholar
  17. [17]
    Showalter R. E.,Monotone Operators in Banach Spaces and Nonlinear Partial Differential Equations, Mat. Surveys Monogr.,49, Amer. Math. Soc., Providence, 1997.Google Scholar

Copyright information

© Springer 2006

Authors and Affiliations

  1. 1.Dipartimento P.A.U.Università degli Studi Mediterranea di Reggio CalabriaReggio CalabriaItaly
  2. 2.Dipartimento P.A.U.Università degli Studi Mediterranea di Reggio CalabriaReggio CalabriaItaly

Personalised recommendations