Equivariant bifurcation index Original Research Article

We consider a bifurcation index View the MathML sourceBIFG(νk0−1)∈U(G) defined in terms of the degree for GG-equivariant gradient maps, see Gȩba (1997) [21], Rybicki (1994) [22], Rybicki (2005) [23], where GG is a real, compact, connected Lie group and U(G)U(G) is the Euler ring of GG, see tom Dieck (1977) [29], tom Dieck (1987) [30]. The main result of this article is the following: View the MathML sourceBIFG(νk0−1)≠Θ∈U(G) iff BIFT(νk0−1)≠Θ∈U(T), Turn MathJax on where T⊂GT⊂G is a maximal torus of GG. It is also shown that all the bifurcation points of weak solutions of the following problem View the MathML source{−Δu=f(u,λ)inBn,u=0onSn−1, Turn MathJax on are global bifurcation points. Additionally, the global symmetry breaking bifurcation points are characterised.