## International Journal of Differential Equations

### Sign-Changing Solutions for Nonlinear Elliptic Problems Depending on Parameters

#### Abstract

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.

#### Article information

Source
Int. J. Differ. Equ., Volume 2010, Special Issue (2010), Article ID 536236, 33 pages.

Dates
Accepted: 23 November 2009
First available in Project Euclid: 26 January 2017

https://projecteuclid.org/euclid.ijde/1485399892

Digital Object Identifier
doi:10.1155/2010/536236

Mathematical Reviews number (MathSciNet)
MR2592740

Zentralblatt MATH identifier
1207.35287

#### Citation

Carl, Siegfried; Motreanu, Dumitru. Sign-Changing Solutions for Nonlinear Elliptic Problems Depending on Parameters. Int. J. Differ. Equ. 2010, Special Issue (2010), Article ID 536236, 33 pages. doi:10.1155/2010/536236. https://projecteuclid.org/euclid.ijde/1485399892

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