*(English)*Zbl 0878.49009

The authors study a new type of eigenvalue problems for variational inequalities. In particular, they study the problem to find $u\in V$ and $\lambda \in \mathbb{R}$ which solve the problem

where $V\subset {L}^{2}\left({\Omega}\right)$ is a real Hilbert space, $a(\xb7,\xb7)$ is a continuous bilinear form and ${j}^{0}(x,u\left(x\right);v\left(x\right))$ is the generalized directional derivative of a locally Lipschitz continuous function $j(x,\xb7)$ at $u\left(x\right)$ along the direction $v\left(x\right)$. 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 $j$ describing the nonlinear part. The second result treats the locally Lipschitz case for $j$ by imposing a more general growth condition.

Finally, the authors illustrate their results by two applications in nonsmooth mechanics.