Bifurcations from degenerate orbits of solutions of nonlinear elliptic systems

TL;DR

This paper investigates global bifurcations of non-constant solutions in nonlinear elliptic systems on spheres and balls, focusing on degenerate and symmetric critical orbits without requiring isolation.

Contribution

It introduces a method to analyze bifurcations from degenerate critical orbits with symmetries, using an index based on equivariant degree theory.

Findings

Established a bifurcation index for degenerate critical orbits.

Extended bifurcation analysis to systems with symmetry groups.

Provided a framework for studying non-isolated critical points.

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.