Multiple and sign-changing solutions for the multivalued p-Laplacian equation

We consider a class of elliptic inclusions under Dirichlet boundary conditions involving multifunctions of Clarke's generalized gradient. Under conditions given in terms of the rst eigenvalue as well as the Fucik spectrum of the p-Laplacian we prove the existence of a positive, a negative and a sign-changing solution. Our approach is based on variational methods for nonsmooth functionals (nonsmooth critical point theory, second deformation lemma), and comparison principles for multivalued elliptic problems. In particular, the existence of extremal constant-sign solutions plays a key role in the proof of sign-changing solutions. (C) 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim