Abstract :
We consider the Dirichlet boundary value problem for an elliptic inclusion governed by a quasilinear
elliptic operator of Leray–Lions type and a multivalued term which is given by the difference
of Clarke’s generalized gradient of some locally Lipschitz function and the subdifferential of some
convex function. Problems of this kind arise, e.g., in mechanical models described by nonconvex
and nonsmooth energy functionals that result from nonmonotone, multivalued constitutive laws. Our
main goal is to characterize the solution set of the problem under consideration. In particular we are
going to prove that the solution set possesses extremal elements with respect to the underlying natural
partial ordering of functions, and that the solution set is compact. The main tools used in the proofs
are abstract results on pseudomonotone operators, truncation, and special test function techniques,
Zorn’s lemma as well as tools from nonsmooth analysis.
2003 Elsevier Inc. All rights reserved.
