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Constant Sign and Nodal Solutions for Problems with the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F820237/MediaObjects/13661_2008_Article_880_IEq1_HTML.gif -Laplacian and a Nonsmooth Potential Using Variational Techniques

  • Ravi P Agarwal1Email author,
  • Michael E Filippakis2,
  • Donal O'Regan3 and
  • Nikolaos S Papageorgiou4
Boundary Value Problems20092009:820237

DOI: 10.1155/2009/820237

Received: 10 December 2008

Accepted: 23 January 2009

Published: 3 March 2009

Abstract

We consider a nonlinear elliptic equation driven by the https://static-content.springer.com/image/art%3A10.1155%2F2009%2F820237/MediaObjects/13661_2008_Article_880_IEq2_HTML.gif -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 https://static-content.springer.com/image/art%3A10.1155%2F2009%2F820237/MediaObjects/13661_2008_Article_880_IEq3_HTML.gif -linear problems.

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Authors’ Affiliations

(1)
Department of Mathematical Sciences, Florida Institute of Technology
(2)
Department of Mathematics, Hellenic Army Academy
(3)
Department of Mathematics, National University of Ireland
(4)
Department of Mathematics, National Technical University

Copyright

© 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.