Equivariant bifurcation index (Q992812)

A bifurcation index \({\mathcal B}l{\mathcal F}_G(v^{-1}_{k_0})\in U(G)\) defined in terms of the degree for \(G\)-equivariant gradient maps, where \(G\) is real, compact, connected Lie group and \(U(G)\) is the Euler ring of \(G\), is considered. The main result is the following: \[ {\mathcal B}l{\mathcal F}_G(v^{-1}_{k_0})\neq \Theta\in U(G)\text{ iff }{\mathcal B}l{\mathcal F}_T(v^{-1}_{k_0})\neq\Theta\in U(T), \] where \(T\subset G\) is a maximal torus of \(G\). Applications of global bifurcations of weak solutions of elliptic differential equations are considered.