Skip to main content
SpringerLink
Log in

Multiple Positive Solutions for Eigenvalue Problems of Hemivariational Inequalities

  • Published:
Positivity Aims and scope Submit manuscript

Abstract

We study a nonlinear eigenvalue problem with a nonsmooth potential. The subgradients of the potential are only positive near the origin (from above) and near +∞. Also the subdifferential is not necessarily monotone (i.e. the potential is not convex). Using variational techniques and the method of upper and lower solutions, we establish the existence of at least two strictly positive smooth solutions for all the parameters in an interval. Our approach uses the nonsmooth critical point theory for locally Lipschitz functions. A byproduct of our analysis is a generalization of a result of Brezis-Nirenberg (CRAS, 317 (1993)) on H10 versus C10 minimizers of a C1-functional.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. W. Allegretto, P. Nistri, P. Zecca, Positive Solutions of Elliptic Nonpositive Problems, Diff. Integral Eqns., 5 (1992), 95–101.

  2. A. Anane, Etude des Valeurs Propres et de la Resonance pour l'Operateur p-Laplacian, Thése de Doctorat, Univesite Libre de Bruxelles (1987).

  3. H. Brezis, L. Nirenberg, H1 versus C1 Local Minimizers, CRAS Paris, 317 (1992), 465–472.

  4. N.P. Cac, A.M. Fink, J.A. Gatica, Nonnegative Solutions of the Radial Laplacian with Nonlinearity that Change Sign, Proc. AMS, 123 (1995), 1393–1398.

  5. A. Castro, R. Shivaji, Nonnegative Solutions for a Class of Nonpositive Problems, Proc. Royal Soc. Edinburgh, 108A (1988), 291–302.

  6. A. Castro, R. Shivaji, Nonnegative Solutions for a Class of Radially Symmetric Nonpositive Problems, Proc. AMS, 106 (1989), 735–740.

  7. A. Castro, R. Shivaji, Positive Solutions for a Concave Semipositone Dirichlet Problems, Nonlin. Anal., 31 (1998), 91–98.

  8. K.C. Chang, Variational Methods for Nondifferentiable Functionals and their Applications to Partial Differential Equations, J. Math. Anal. Appl., 80 (1981), 102–129.

  9. F.H. Clarke, A New Approach to Lagrange Multipliers, Math. Oper. Research, 1 (1976), 165–173.

  10. F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York (1983).

  11. D.S. Cohen, H.B. Keller, Some Positive Problems Suggested by Nonlinear Heat Generation, J. Math. Mech., 16 (1967), 1361–1376.

    Google Scholar 

  12. Z. Denkowski, S. Migórski, N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Theory, Kluwer/Plenum, New York (2003).

  13. Z. Denkowski, S. Migórski, N.S. Papageorgiou, An Introduction to Nonlinear Analysis: Applications, Kluwer/Plenum, New York (2003).

  14. A. Dontchev, J. Zolezzi, Well-Posed Optimization Problems, Lecture Notes in Math., Vol. 1543, Springer-Verlag, Berlin (1993).

  15. D.D. Hai, Positive Solutions to a Class of Elliptic Boundary Value Problems, J. Math. Anal. Appl., 227 (1998), 195–199.

    Google Scholar 

  16. S. Hu, N.S. Papageorgiou, Handbook of Multivalued Analysis. Volume I: Theory, Kluwer, Dordrecht, The Netherlands (1997).

  17. S. Hu, N.S. Papageorgiou, Handbook of Multivalued Analysis. Volume II: Applications, Kluwer, Dordrecht, The Netherlands (2000).

  18. N. Kourogenis, N.S. Papageorgiou, Nonsmooth Critical Point Theory and Nonlinear Elliptic Equations at Resonance, J. Austr. Math. Soc., 69A (2000), 245–271.

  19. A. Kufner, O. John, S. Fucik, Functional Spaces, Noordhoff International Publishing, Leyden, The Netherlands (1977).

  20. P.L. Lions On the Existence of Positive Solutions of Semilinear Elliptic Equations, SIAM Review 24 (1982), 441–467.

    Google Scholar 

  21. D. Motreanu, Eigenvalue Problems for Variational-Hemivariational Inequalities in the Sense of P.D. Panagiotopoulos, Nonlin. Anal., 47 (2001), 5101–5112.

  22. Z. Naniewicz, P.D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications, Marcel-Dekker, New York (1995).

  23. M. Struwe, Variational Methods, Springer-Verlag, Berlin (1990).

  24. J.L. Vasquez, A Strong Maximum Principle for Some Quasilinear Elliptic Equations, Appl. Math. Optim., 12 (1984), 191–202.

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, National Technical University, Zografou Campus, Athens, 15780, Greece

    Michael Filippakis & Nikolaos S. Papageorgiou

  2. Institute of Computer Science, Jagiellonian University, ul. Nawojki 11, 30072, Cracow, Poland

    Leszek Gasiński

Authors
  1. Michael Filippakis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Leszek Gasiński
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Nikolaos S. Papageorgiou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nikolaos S. Papageorgiou.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Filippakis, M., Gasiński, L. & Papageorgiou, N. Multiple Positive Solutions for Eigenvalue Problems of Hemivariational Inequalities. Positivity 10, 491–515 (2006). https://doi.org/10.1007/s11117-005-0002-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11117-005-0002-5

Mathematics Subject Classification (2000)

Keywords

Search

Navigation