On the eigenvalue problem for hemivariational inequalities: Existence and multiplicity of solutions
The paper deals with the existence of solutions and the multiplicity of solutions for hemivariational inequalities of the form \[ a(u,v)- \langle f,v\rangle_V+\int_\Omega j^0 (x,u;v)dx\geq \lambda(u,v)_V\quad \forall v\in V, \] where \(u\in S_r:= \{v\in V:|v|_V=r\}\) and \(\lambda\in R\) are the basic unknowns. The novelties of the presented approach in comparison with the classical results of the critical point theory are the following: 1. The eigensolution \(u\) is assumed to lie in the prescribed sphere \(S_r\); 2. The mountain-pass theorem of Ambrosetti and Rabinowitz is not applicable due to the existence of a linear part \(\langle f,\cdot\rangle_V\); 3. \(V\) is a vector-valued function space which does not allow \(j\) to be of the form of a primitive. The main results of the paper establish both the existence and multiplicity of solutions. In particular, the conditions ensuring a countable set of solutions for hemivariational inequalities under consideration have been derived. One can learn from the paper how to efficiently extend and modify the methods of the classical critical point theory to the nonsmooth case represented by hemivariational inequalities involving locally Lipschitz functionals. Interesting examples of mechanical problems concerning the behavior of a plane elastic body subject to nonmonotone possibly multivalued skin friction conditions and the behavior of two adhesively connected plates illustrate the theoretical results. The paper is very interesting not only for those who deal with hemivariational inequalities but also for readers working in the field of nonsmooth analysis.
scientific article; zbMATH DE number 902428