Bifurcation theory for symmetric potential operators and the equivariant cup-length
Let G be a compact Lie group, E a Hilbert space on which G acts orthogonally and \(\phi: {\mathbb{R}}\times E\to {\mathbb{R}}\) a \(C^ 2\)-functional which is invariant with respect to the action of G on E. Assume that \(0\in E\) is a critical point of \(\phi(\lambda\),.) for all \(\lambda\in {\mathbb{R}}\). We are interested in critical points of \(\phi(\lambda\),.) for \(\lambda\) near \(\lambda_ 0\) which bifurcate from \((\lambda_ 0,0)\). We give lower bounds for the number of bifurcating G-orbits of critical points. For \(G={\mathbb{Z}}/2\) this problem has been considered e.g. by \textit{E. R. Fadell} and \textit{P. H. Rabinowitz} [J. Funct. Anal. 26, 48-67 (1977; Zbl 0363.47029)]. We generalize their result in that we allow more general potentials and by replacing \({\mathbb{Z}}/2\) by an arbitrary compact Lie group. The main result about bifurcation is reduced to a generalized mountain pass theorem. The proof uses a new index theory based on the concept of cup length in the equivariant context.
scientific article; zbMATH DE number 4117518
scientific article