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Zbl 0878.49009
Goeleven, D.; Motreanu, D.; Panagiotopoulos, P.D.
Multiple solutions for a class of eigenvalue problems in hemivariational inequalities.
(English)
[J] Nonlinear Anal., Theory Methods Appl. 29, No.1, 9-26 (1997). ISSN 0362-546X

The authors study a new type of eigenvalue problems for variational inequalities. In particular, they study the problem to find $u\in V$ and $\lambda\in\bbfR$ which solve the problem $$a(u,v)-\lambda \int_\Omega u\cdot vdx+ \int_\Omega j^0(x,u;v)dx\quad\text{for all }v\in V,\tag P$$ where $V\subset L^2(\Omega)$ is a real Hilbert space, $a(\cdot,\cdot)$ is a continuous bilinear form and $j^0(x,u(x);v(x))$ is the generalized directional derivative of a locally Lipschitz continuous function $j(x,\cdot)$ 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.\par 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.\par 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.\par Finally, the authors illustrate their results by two applications in nonsmooth mechanics.
[A.Masiello (Bari)]
MSC 2000:
*49J40 Variational methods including variational inequalities
58E05 Abstract critical point theory

Keywords: hemivariational inequalities; model of hysteresis; nonsmooth critical point theory; eigenvalue problems; variational inequalities; nonconvex functionals; nonsmooth mechanics

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