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  • Research Article
  • Open Access

Discontinuous Variational-Hemivariational Inequalities Involving the -Laplacian

Journal of Inequalities and Applications20072007:013579

  • Received: 6 August 2007
  • Accepted: 25 November 2007
  • Published:


We deal with discontinuous quasilinear elliptic variational-hemivariational inequalities. By using the method of sub- and supersolutions and based on the results of S. Carl, we extend the theory for discontinuous problems. The proof of the existence of extremal solutions within a given order interval of sub- and supersolutions is the main goal of this paper. In the last part, we give an example of the construction of sub- and supersolutions.


  • Extremal Solution
  • Order Interval
  • Discontinuous Problem


Authors’ Affiliations

Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, Halle, 06099, Germany


  1. Clarke FH: Optimization and Nonsmooth Analysis, Classics in Applied Mathematics. Volume 5. 2nd edition. SIAM, Philadelphia, Pa, USA; 1990:xii+308.View ArticleGoogle Scholar
  2. Carl S: Existence and comparison results for variational-hemivariational inequalities. Journal of Inequalities and Applications 2005, (1):33–40.Google Scholar
  3. Carl S, Le VK, Motreanu D: Nonsmooth Variational Problems and Their Inequalities, Springer Monographs in Mathematics. Springer, New York, NY, USA; 2007:xii+395.View ArticleGoogle Scholar
  4. Appell J, Zabrejko PP: Nonlinear Superposition Operators, Cambridge Tracts in Mathematics. Volume 95. Cambridge University Press, Cambridge, UK; 1990:viii+311.View ArticleGoogle Scholar
  5. Zeidler E: Nonlinear Functional Analysis and Its Applications Volume II/B. Springer, Berlin, Germany; 1990.View ArticleGoogle Scholar
  6. Heinonen J, Kilpeläinen T, Martio O: Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, NY, USA; 1993:vi+363.Google Scholar
  7. Carl S, Heikkilä S: Nonlinear Differential Equations in Ordered Spaces. Chapman & Hall/CRC, Boca Raton, Fla, USA; 2000:vi+323.MATHGoogle Scholar
  8. Anane A: Simplicité et isolation de la première valeur propre du-laplacien avec poids. Comptes Rendus des Séances de l'Académie des Sciences. Série I. Mathématique 1987,305(16):725–728.MathSciNetMATHGoogle Scholar
  9. Carl S, Motreanu D: Sign-changing and extremal constant-sign solutions of nonlinear elliptic problems with supercritical nonlinearities. Communications on Applied Nonlinear Analysis 2007,14(4):85–100.MathSciNetMATHGoogle Scholar


© Patrick Winkert 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.