Skip to content


Open Access

Constant Sign and Nodal Solutions for Problems with the -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

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.


Differential EquationPartial Differential EquationOrdinary Differential EquationFunctional EquationFull Article

Publisher note

To access the full article, please see PDF.

Authors’ Affiliations

Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, USA
Department of Mathematics, Hellenic Army Academy, Vari, Greece
Department of Mathematics, National University of Ireland, Galway, Ireland
Department of Mathematics, National Technical University, Athens, Greece


© 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.