Bifurcation of solutions of $U(1)$-equivariant semilinear boundary value problems

Assuming that there is a known (trivial) branch of solutions of a parameterized family of equations, topological bifurcation studies the topological invariants of the linearized equations along the trivial branch whose nonvanishing entails the appearance of bifurcation from the trivial branch. We introduce here some refined topological invariants for semilinear elliptic boundary value problems equivariant with respect to the action of the circle $U(1)$ allowing to improve, in this case, some previously obtained bifurcation criteria for general nonlinear elliptic boundary value problems.