Bifurcations from degenerate orbits of solutions of nonlinear elliptic systems

Abstract

The aim of this paper is to study global bifurcations of non-constant solutions of some nonlinear elliptic systems, namely the system on a sphere and the Neumann problem on a ball. We study the bifurcation phenomenon from families of constant solutions given by critical points of the potentials. Considering this problem in the presence of additional symmetries of a compact Lie group, we study orbits of solutions and, in particular, we do not require the critical points to be isolated. Moreover, we allow the considered orbits of critical points to be degenerate. To prove the bifurcation, we compute the index of an isolated degenerate critical orbit in an abstract situation. This index is given in terms of the degree for equivariant gradient maps.