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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 uV and λ which solve the problem

a(u,v)-λ Ω u·vdx+ Ω j 0 (x,u;v)dxforallvV,(P)

where VL 2 (Ω) is a real Hilbert space, a(·,·) is a continuous bilinear form and j 0 (x,u(x);v(x)) is the generalized directional derivative of a locally Lipschitz continuous function j(x,·) at u(x) along the direction v(x). 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.

49J40Variational methods including variational inequalities
58E05Abstract critical point theory