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 -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 -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 -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.