A $K$-theoretical invariant and bifurcation for a parameterized family of functionals

TL;DR

This paper introduces a $K$-theoretical invariant to analyze bifurcations in a parameterized family of functionals on Hilbert manifolds, providing conditions for when bifurcations occur from the trivial solution.

Contribution

It develops a new $K$-theoretical approach to detect bifurcations in families of functionals depending on parameters, extending previous methods.

Findings

Established a sufficient condition for bifurcation based on the $K$-theoretical invariant.

Applied the invariant to a family of functionals on Hilbert manifolds.

Provided a framework for analyzing bifurcations in infinite-dimensional settings.

Abstract

For a family of functionals defined on a Hilbert manifold and smoothly depending on a compact finite dimensional manifold, we give a sufficient condition on the parameter space in such a way the family bifurcate from the trivial branch.