Global Bifurcations of Solutions of Elliptic Differential Equations

The aim of this article is to prove global bifurcation theorems for S1-equivariant potential operators of the form ‘‘compact perturbation of identity.’’ As an application we prove that components of the set of nontrivial solutions of system yDu x.sl? =f u x.. in V us0 on ­V, which bifurcate from the set of trivial solutions are unbounded in suitably chosen space. The important point to note here is that our results apply only to systems with an even number of equations. As the main tool we use degree theory for S1-equivariant potential operators constructed by the author in w16x and developed in w17x.