Location of Solutions for General Nonsmooth Problems

  • D. Motreanu
  • V. Rădulescu
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 67)


The aim of the present Chapter is to study from a qualitative point of view a general eigenvalue problem associated to a variational-hemivariational inequality with a constraint for the eigenvalue. The basic feature of our approach is that we are mainly concerned with the location of eigensolution (u, λ), where u and λ stand for the eigenfunction and the eigenvalue, respectively. This is done in Section 2, where the location of eigensolutions is achieved by means of the graph of the derivative of a C 1 function. Section 1 presents a general existence result for variationalhemivariational inequalities with assumptions of Ambrosetti and Rabinowitz type. Section 2 deals with the exposition of our abstract location results. In Section 3 we discuss the location of solutions to variationalhemivariational inequalities by applying the abstract results. The case of nonlinear Dirichlet boundary value problems is contained.


Eigenvalue Problem Variational Method Generalize Gradient Real Hilbert Space Critical Point Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    S. Adly and D. Motreanu, Location of eigensolutions to variationalhemivariational inequalities, J. Nonlinear Convex Anal. 1 (2000), 255–270.MathSciNetzbMATHGoogle Scholar
  2. [2]
    A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Func. Anal. 14 (1973), 349–381.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [3]
    C. Baiocchi and A. Capelo, Variational and Quasivariational Inequalities. Applications to Free Boundary Problems, John Wiley and Sons, New York, 1984.zbMATHGoogle Scholar
  4. [4]
    H. Brézis and L. Nirenberg, Functional Analysis and Applications to Partial Differential Equations, in preparation.Google Scholar
  5. [5]
    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
  6. [6]
    F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley & Sons, New York (1983).zbMATHGoogle Scholar
  7. [7]
    S. Dâbuleanu and D. Motreanu, Existence results for a class of eigenvalue quasi-linear problems with nonlinear boundary condition, Adv. Nonlinear Var. In-equal. 2 (1999), 41–54.zbMATHGoogle Scholar
  8. [8]
    G. Dincâ, P. Jebelean and D. Motreanu, Existence and approximation for a general class of differential inclusions, Houston J. Math. 28 (2002), 193–215.zbMATHGoogle Scholar
  9. [9]
    D. Goeleven, D. Motreanu, Y. Dumont and M. Rochdi, Variational and Hemivariational Inequalities, Theory, Methods and Applications, Volume I: Unilateral Analysis and Unilateral Mechanics, Kluwer Academic Publishers, Dordrecht/Boston/London, to appear.Google Scholar
  10. [10]
    D. Goeleven and D. Motreanu, Variational and Hemivariational Inequalities, Theory, Methods and Applications, Volume II: Unilateral Problems, Kluwer Academic Publishers, Dordrecht/Boston/London, to appear.Google Scholar
  11. [11]
    J. Haslinger, M. Miettinen and P. D. Panagiotopoulos, Finite Element Method for Hemivariational Inequalities. Theory, Methods and Applications, Kluwer Academic Publishers, Nonconvex Optimization and Its Applications, Vol. 35, Dordrecht/Boston/London, 1999.Google Scholar
  12. [12]
    J. Haslinger and D. Motreanu, Hemivariational inequalities with a general growth condition: existence and approximation, Appl. Anal., to appear.Google Scholar
  13. [13]
    P. Mironescu and V. Râdulescu, The study of a bifurcation problem associated to an asymptotically linear function, Nonlinear Anal. 26 (1996), 857–875.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [14]
    E. S. Mistakidis and G. E. Stavroulakis, Nonconvex Optimization in Mechanics, Nonconvex Optimization and Its Applications, Vol. 21, Kluwer Academic Publishers, Dordrecht/Boston/London, 1998.Google Scholar
  15. [15]
    D. Motreanu, A saddle-point approach to nonlinear eigenvalues problems, Math. Slovaca 47 (1997), 463–477.MathSciNetzbMATHGoogle Scholar
  16. [16]
    D. Motreanu, Eigenvalue problems for variational-hemivariational inequalities in the sense of P. D. Panagiotopoulos, Nonlinear Anal. 47 (2001), 5101–5112.MathSciNetzbMATHGoogle Scholar
  17. [17]
    D. Motreanu: Nonsmooth/Nonconvex Mechanics: Modeling, Analysis and Numerical Methods, A Volume dedicated to the memory of Professor P.D. Panagiotopoulos, D. Gao, R. W. Ogden and G. E. Stavroulakis (eds.), Kluwer Academic Publishers, Dordrecht, Boston, London, 2001.Google Scholar
  18. [18]
    D. Motreanu, Existence and multiplicity results for variational-hemivariational inequalities in the sense of P. D. Panagiotopoulos, in: Proceedings of the International Conference on Nonsmooth/Nonconvex Mechanics with Applications in Engineering, In memoriam of Professor P. D. Panagiotopoulos, 5–6 July 2002, Thessaloniki, Greece, pp. 23–30.Google Scholar
  19. [19]
    D. Motreanu and P. D. Panagiotopoulos, Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities and Applications, Kluwer Academic Publishers, Nonconvex Optimization and Its Applications, Vol. 29, Dordrecht/Boston/London, 1999.Google Scholar
  20. [20]
    D. Motreanu and V. Radulescu, Existence theorems for some classes of boundary value problems involving the p-Laplacian, Panam. Math. J. 7 (1997), 53–66.MathSciNetzbMATHGoogle Scholar
  21. [21]
    Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel Dekker, Inc., New York (1995).Google Scholar
  22. [22]
    P. D. Panagiotopoulos, Inequality Problems in Mechanics and Applications. Convex and Nonconvex Energy Functions, Birkhäuser Verlag, Basel, 1985.zbMATHCrossRefGoogle Scholar
  23. [23]
    P. D. Panagiotopoulos, Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer-Verlag, Berlin, New York, 1993.Google Scholar
  24. [24]
    P.H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. 65, Amer. Math. Soc., Providence, R.I., 1996.Google 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

Personalised recommendations