Equivariant bifurcation index

Abstract

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