Constant Sign and Nodal Solutions for Problems with the -Laplacian and a Nonsmooth Potential Using Variational Techniques

  • Ravi P Agarwal1Email author,

    Affiliated with

    • Michael E Filippakis2,

      Affiliated with

      • Donal O'Regan3 and

        Affiliated with

        • Nikolaos S Papageorgiou4

          Affiliated with

          Boundary Value Problems20092009:820237

          DOI: 10.1155/2009/820237

          Received: 10 December 2008

          Accepted: 23 January 2009

          Published: 3 March 2009


          We consider a nonlinear elliptic equation driven by the -Laplacian with a nonsmooth potential (hemivariational inequality) and Dirichlet boundary condition. Using a variational approach based on nonsmooth critical point theory together with the method of upper and lower solutions, we prove the existence of at least three nontrivial smooth solutions: one positive, the second negative, and the third sign changing (nodal solution). Our hypotheses on the nonsmooth potential incorporate in our framework of analysis the so-called asymptotically -linear problems.

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          Authors’ Affiliations

          Department of Mathematical Sciences, Florida Institute of Technology
          Department of Mathematics, Hellenic Army Academy
          Department of Mathematics, National University of Ireland
          Department of Mathematics, National Technical University


          © Ravi P. Agarwal et al. 2009

          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.