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International Journal of Differential Equations
Volume 2010, Article ID 536236, 33 pages
Research Article

Sign-Changing Solutions for Nonlinear Elliptic Problems Depending on Parameters

1Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle, Germany
2Département de Mathématiques, Université de Perpignan, 66860 Perpignan, France

Received 18 September 2009; Accepted 23 November 2009

Academic Editor: Thomas Bartsch

Copyright © 2010 Siegfried Carl and Dumitru Motreanu. 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.


The study of multiple solutions for quasilinear elliptic problems under Dirichlet or nonlinear Neumann type boundary conditions has received much attention over the last decades. The main goal of this paper is to present multiple solutions results for elliptic inclusions of Clarke's gradient type under Dirichlet boundary condition involving the 𝑝 -Laplacian which, in general, depend on two parameters. Assuming different structure and smoothness assumptions on the nonlinearities generating the multivalued term, we prove the existence of multiple constant-sign and sign-changing (nodal) solutions for parameters specified in terms of the Fučik spectrum of the 𝑝 -Laplacian. Our approach will be based on truncation techniques and comparison principles (sub-supersolution method) for elliptic inclusions combined with variational and topological arguments for, in general, nonsmooth functionals, such as, critical point theory, Mountain Pass Theorem, Second Deformation Lemma, and the variational characterization of the “beginning”of the Fu\v cik spectrum of the 𝑝 -Laplacian. In particular, the existence of extremal constant-sign solutions and their variational characterization as global (resp., local) minima of the associated energy functional will play a key-role in the proof of sign-changing solutions.