Open Access
Translator Disclaimer
2010 Sign-Changing Solutions for Nonlinear Elliptic Problems Depending on Parameters
Siegfried Carl, Dumitru Motreanu
Int. J. Differ. Equ. 2010(SI3): 1-33 (2010). DOI: 10.1155/2010/536236


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 Fučik 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.


Download Citation

Siegfried Carl. Dumitru Motreanu. "Sign-Changing Solutions for Nonlinear Elliptic Problems Depending on Parameters." Int. J. Differ. Equ. 2010 (SI3) 1 - 33, 2010.


Received: 18 September 2009; Accepted: 23 November 2009; Published: 2010
First available in Project Euclid: 26 January 2017

zbMATH: 1207.35287
MathSciNet: MR2592740
Digital Object Identifier: 10.1155/2010/536236

Rights: Copyright © 2010 Hindawi


Vol.2010 • No. SI3 • 2010
Back to Top