This Chapter deals with general techniques for studying the existence and multiplicity of critical points of nondifferentiable functionals in the so-called limit case (see Remark 3.1). There are proved nonsmooth versions of several celebrated results like: Deformation Lemma, Mountain Pass Theorem, Saddle Point Theorem, Generalized Mountain Pass Theorem. First, we present a general deformation result for nonsmooth functionals which can be expressed as a sum of a locally Lipschitz function and a concave, proper, upper semicontinuous function. Then we give a general minimax principle for nonsmooth functionals which can be expressed as a sum of a locally Lipschitz function and a convex, proper, lower semicontinuous functional. Here we are concerned with the limit case (i.e. the equality c = a, see Remark 3.1), obtaining results which are complementary to the minimax principles in Section 2 of Chapter 2. These general results are applied in the second Section of this Chapter for proving existence, multiplicity and location of solutions to various boundary value and unilateral problems with discontinuous nonlinearities.


Variational Method Critical Point Theory Hemivariational Inequality Finite Covering Mountain Pass Theorem 
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  1. [1]
    A. Ambrosetti, Critical points and nonlinear variational problems, Mem. Soc. Math. France (N.S.) 49, 1992.Google Scholar
  2. [2]
    A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349–381.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    H. Brézis, Analyse Fonctionnelle - Théorie et Applications, Masson, Paris, 1983.zbMATHGoogle Scholar
  4. [4]
    K.-C. Chang, Variational methods for non-differentiable functionals and their applications to partial differential equations, J. Math. Anal. Appl. 80 (1981), 102–129.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [5]
    F. H. Clarke, Optimization and Nonsmooth Analysis, Classics Appl. Math. 5, SIAM, Philadelphia, 1990.Google Scholar
  6. [6]
    Y. Du, A deformation lemma and some critical point theorems, Bull. Austral. Math. Soc. 43 (1991), 161–168.MathSciNetzbMATHCrossRefGoogle Scholar
  7. [7]
    J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.zbMATHGoogle Scholar
  8. [8]
    L. Gasinski and N. S. Papageorgiou, Solutions and multiple solutions for quasi-linear hemivariational inequalities at resonance, Froc. Royal Soc. Edinburgh (Math) 131A (2001), 1091–1111.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [9]
    N. Ghoussoub and D. Preiss, A general mountain pass principle for locating and classifying critical points, Ann. Inst. Henri Poincaré Anal. Non Linéaire 6 (1989), 321–330.MathSciNetzbMATHGoogle Scholar
  10. [10]
    D. Goeleven, D. Motreanu and P. D. Panagiotopoulos, Eigenvalue problems for variational-hemivariational inequalities at resonance, Nonlinear Anal. 33 (1998), 161–180.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [11]
    S. Marano and D. Motreanu, Infinitely many critical points of non-differentiable functions and applications to a Neumann type problem involving the p-Laplacian, J. Differ. Equations 182 (2002), 108–120.MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    S. Marano and D. Motreanu, A deformation theorem and some critical point results for non-differentiable functions, submitted.Google Scholar
  13. [13]
    D. Motreanu and V. V. Motreanu, Duality in nonsmooth critical point theory, limit case and applications, submitted.Google Scholar
  14. [14]
    D. Motreanu and P. D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities, Nonconvex Optimization and its Applications 29, Kluwer Academic Publishers, Dordrecht/Boston/London, 1998.Google Scholar
  15. [15]
    D. Motreanu and C. Varga, Some critical point results for locally Lipschitz functionals, Comm. Appl. Nonlinear Anal. 4 (1997), 17–33.MathSciNetzbMATHGoogle Scholar
  16. [16]
    P. D. Panagiotopoulos, Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993.zbMATHCrossRefGoogle Scholar
  17. [17]
    P. Pucci and J. Serrin, A mountain pass theorem, J. Differ. Equations 60 (1985), 142–149.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [18]
    P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Reg. Conf. Ser. in Math. 65, Amer. Math. Soc., Providence, 1986.Google Scholar
  19. [19]
    P. H. Rabinowitz, Sonic aspects of critical point theory, in: Proceedings of the 1982 Changchun Symposium on Differential Geometry and Differential Equations (S.S. Chern, R. Wang and M. Chi (eds)), Science Press, Bijing, 1986, pp. 185–232.Google Scholar
  20. [20]
    A. Szulkin, Minimax principles for lower semicontinuous functions and applications to nonlinear boundary value problems, Ann. Inst. Henri Poincaré. Anal. Non Linéaire 3 (1986), 77–109.MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2003

Authors and Affiliations

  • D. Motreanu
    • 1
  • V. Rădulescu
    • 2
  1. 1.Department of MathematicsUniversity of PerpignanPerpignanFrance
  2. 2.Department of MathematicsUniversity of CraiovaCraiovaRomania

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