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Multiple solutions for a class of eigenvalue problems in hemivariational inequalities. (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 $\lambda \in ℝ$ which solve the problem

$a\left(u,v\right)-\lambda {\int }_{{\Omega }}u·vdx+{\int }_{{\Omega }}{j}^{0}\left(x,u;v\right)dx\phantom{\rule{1.em}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}v\in V,\phantom{\rule{2.em}{0ex}}\left(\mathrm{P}\right)$

where $V\subset {L}^{2}\left({\Omega }\right)$ is a real Hilbert space, $a\left(·,·\right)$ is a continuous bilinear form and ${j}^{0}\left(x,u\left(x\right);v\left(x\right)\right)$ is the generalized directional derivative of a locally Lipschitz continuous function $j\left(x,·\right)$ 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.

##### MSC:
 49J40 Variational methods including variational inequalities 58E05 Abstract critical point theory