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This article is part of the series Singular Boundary Value Problems for Ordinary Differential Equations.

Open Access Open Badges Research Article

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

RaviP Agarwal1*, MichaelE Filippakis2, Donal O'Regan3 and NikolaosS Papageorgiou4

Author Affiliations

1 Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901-6975, USA

2 Department of Mathematics, Hellenic Army Academy, Vari, 16673 Athens, Greece

3 Department of Mathematics, National University of Ireland, Galway, Ireland

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

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Boundary Value Problems 2009, 2009:820237  doi:10.1155/2009/820237

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2009/1/820237

Received:10 December 2008
Revisions received:21 January 2009
Accepted:23 January 2009
Published:3 March 2009

© 2009 The Author(s).

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.


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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