Connected sets of solutions of symmetric elliptic systems

TL;DR

This paper investigates bifurcations of connected solution sets in symmetric elliptic systems using equivariant bifurcation theory, and applies these results to analyze nonconstant solutions of a nonlinear Neumann problem.

Contribution

It introduces a Rabinowitz type alternative for symmetric gradient operators and applies it to nonlinear Neumann problems, advancing bifurcation analysis in symmetric elliptic systems.

Findings

Established a Rabinowitz type bifurcation alternative for symmetric gradient operators.

Applied abstract bifurcation results to nonlinear Neumann boundary value problems.

Identified conditions for bifurcations of connected sets of solutions in symmetric elliptic systems.

Abstract

The purpose of this paper is twofold. First we study bifurcations of connected sets of critical orbits of some invariant functional from a given family of critical orbits. We use techniques of equivariant bifurcation theory to obtain a Rabinowitz type alternative for symmetric gradient operators. The second aim is to apply the abstract results to studying orbits of nonconstant solutions of a nonlinear Neumann problem.