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Bifurcation of stationary solutions to quasivariational inequalities. (English) Zbl 0808.35039

A bifurcation problem for quasivariational inequalities of the type \[ u\in K(u),\qquad (\lambda u-Au -G(\lambda, u),v- u)\geq 0 \quad \forall v\in K(u) \tag \(*\) \] is considered. Here \(\{K(u)\); \(u\in H\}\) is a system of closed convex sets in the real Hilbert space \(H\) satisfying certain assumptions, \(A: H\to H\) is a linear completely continuous operator, \(G: \mathbb{R}\times H\to H\) a small compact perturbation, \(\lambda\) a real bifurcation parameter. The existence of bifurcation points of \((*)\) lying in intervals \((\lambda_ 1, \lambda_ 2)\) is proved where \(\lambda_ 1\), \(\lambda_ 2\) are eigenvalues of a certain type of the operator \(A\). Moreover, it is shown that under certain assumptions there exists a bifurcation point of \((*)\) greater than the greatest real eigenvalue of the operator \(A\). This can occur even in case of a symmetric operator \(A\). Notice that this is excluded if \(A\) is symmetric and \(K(u) =K\) is a fixed closed convex cone with the vertex at the origin, i.e. if \((*)\) is a standard variational inequality.
Reviewer: M.Kučera (Praha)

MSC:

35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000)
35B32 Bifurcations in context of PDEs
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