Global bifurcations of critical orbits of GG-invariant strongly indefinite functionals

Let HH be a separable Hilbert space which is an orthogonal representation of a compact Lie group GG and let Φ:H→RΦ:H→R be a GG-invariant strongly indefinite functional of the class C1C1. To study critical orbits of ΦΦ we have defined the degree for GG-invariant strongly indefinite functionals, which is an element of the Euler ring U(G)U(G). Using this degree we have formulated the Rabinowitz alternative for GG-invariant strongly indefinite functionals. The abstract results are applied to the study of global bifurcations of weak solutions of non-cooperative systems of elliptic differential equations.