The authors study a new type of eigenvalue problems for variational inequalities. In particular, they study the problem to find and which solve the problem
where is a real Hilbert space, is a continuous bilinear form and is the generalized directional derivative of a locally Lipschitz continuous function at along the direction . Such class of problems can be considered as a generalization of the variational inequalities to nonconvex functionals. They are strongly motivated by various problems in mechanics having a lack of convexity.
In this paper, a min-max approach for even functionals is used to prove multiplicity results for the problem (P). They extend some previous existence results obtained by using a version of the Mountain Pass Theorem.
Two results are proved. The first one concerns essentially the Lipschitz case for the function describing the nonlinear part. The second result treats the locally Lipschitz case for by imposing a more general growth condition.
Finally, the authors illustrate their results by two applications in nonsmooth mechanics.