Bertocchi, Carla; Degiovanni, Marco On the existence of two branches of bifurcation for eigenvalue problems associated with variational inequalities. (English) Zbl 0828.58007 Manara, Carlo Felice (ed.) et al., Papers in honor of Giovanni Melzi. Milano: Univ. Cattolica del Sacro Cuore. Sci. Mat. 11, 35-72 (1994). In the paper the following “variational bifurcation” problem is considered. Let \(H\) be a Hilbert space, \(W\) a convex subset of \(H\), \(f : \overline W \to\;] - \infty, + \infty]\) a function, and \(L : H \to H\) a symmetric linear bounded operator. The bifurcation problem for \(L\) and \(f\) is then \[ \begin{cases} (\lambda,u) \in \mathbb{R} \times W \\ \lambda Lu \in \partial^- f(u) \end{cases} \tag{1} \] where the subdifferential \(\partial^-f\) is appropriately defined. The function \[ f_0 (u) = \Gamma \lim_{\varepsilon \to 0^+} \varepsilon^{-2} f(\varepsilon u) \] is also defined, so that the “linearized problem” is written as \[ \begin{cases} (\lambda,u) \in \mathbb{R} \times H \\ \lambda Lu \in \partial^- f_0 (u). \end{cases} \tag{2} \] It is shown that every bifurcation value of (1) is an eigenvalue of (2). The converse is not true in general; counterexamples can be found in Ann. Mat. Pura Appl., IV. Ser. 156, 37-71 (1990; Zbl 0722.58013) and Ann. Fac. Sci. Toulouse, V. Sér., Math. 11, No. 1, 39-66 (1990; Zbl 0717.49010) by the second author and in Comment. Math. Univ. Carol. 24, No. 4, 657-665 (1983; Zbl 0638.49020) by E. Miersemann. However, the converse holds when \(f_0\) behaves like a quadratic form, and in the paper it is proved that in this case there are at least two branches of bifurcation.In Section 4 an application to variational inequalities is given.For the entire collection see [Zbl 0802.00005]. Reviewer: G.Buttazzo (Pisa) Cited in 3 Documents MSC: 58E07 Variational problems in abstract bifurcation theory in infinite-dimensional spaces 49J45 Methods involving semicontinuity and convergence; relaxation 58E35 Variational inequalities (global problems) in infinite-dimensional spaces 49J40 Variational inequalities Keywords:nonsmooth analysis; \(\Gamma\)-convergence; variational bifurcation; variational inequalities Citations:Zbl 0722.58013; Zbl 0717.49010; Zbl 0638.49020 PDFBibTeX XMLCite \textit{C. Bertocchi} and \textit{M. Degiovanni}, Sci. Mat. 11, 35--72 (1994; Zbl 0828.58007)