Boundary Value Problems
Volume 2010 (2010), Article ID 139126, 22 pages
Research Article

Sign-Changing and Extremal Constant-Sign Solutions of Nonlinear Elliptic Neumann Boundary Value Problems

Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany

Received 23 November 2009; Accepted 15 June 2010

Academic Editor: Pavel Drábek

Copyright © 2010 Patrick Winkert. 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.


Our aim is the study of a class of nonlinear elliptic problems under Neumann conditions involving the p-Laplacian. We prove the existence of at least three nontrivial solutions, which means that we get two extremal constant-sign solutions and one sign-changing solution by using truncation techniques and comparison principles for nonlinear elliptic differential inequalities. We also apply the properties of the Fuc̆ik spectrum of the p-Laplacian and, in particular, we make use of variational and topological tools, for example, critical point theory, Mountain-Pass Theorem, and the Second Deformation Lemma.