Global bifurcation index of critical orbit of strongly indefinite functional

TL;DR

This paper introduces a new index for critical orbits in strongly indefinite functionals, relates it to critical point indices, and applies it to prove bifurcation and existence of solutions in nonlinear elliptic systems.

Contribution

It defines a global bifurcation index for critical orbits and demonstrates its use in analyzing bifurcation phenomena in nonlinear elliptic systems.

Findings

Established a relationship between the orbit index and the restricted critical point index.

Proved bifurcation of nontrivial solutions in nonlinear elliptic systems.

Demonstrated existence of unbounded solution sets.

Abstract

In this paper we study an index of a critical orbit, defined in terms of the degree for invariant strongly indefinite functionals. We establish a relationship of this index with the index of a critical point of the mapping restricted to the space normal to the orbit. The second aim of the article is to use the index of a critical orbit to prove the bifurcation of nontrivial solutions of nonlinear elliptic systems with Neumann boundary conditions. We consider also the existence of unbounded sets of solutions of such systems.