Bifurcations from the orbit of solutions of the Neumann problem

TL;DR

This paper investigates bifurcations from orbits of solutions in a nonlinear Neumann problem on a ball, using equivariant analysis to identify conditions for symmetry-breaking and bifurcation phenomena.

Contribution

It introduces a framework for analyzing bifurcations from non-isolated critical points considering symmetry, extending previous methods to orbit-based solutions.

Findings

Established conditions for local bifurcations.

Identified criteria for global bifurcations.

Characterized symmetry-breaking orbits.

Abstract

The purpose of this paper is to study weak solutions of a nonlinear Neumann problem considered on a ball. Assuming that the potential is invariant, we consider an orbit of critical points, i.e. we do not assume that critical points are isolated. We apply techniques of the equivariant analysis to examine bifurcations from the orbits of trivial solutions. We formulate sufficient conditions for local and global bifurcations, in terms of the right-hand side of the system and eigenvalues of the Laplace operator. Moreover, we characterise orbits at which the global symmetry-breaking phenomenon occurs.