On the Minimax Bifurcation Formula

TL;DR

This paper introduces a variational minimax method for directly detecting maximal saddle-node bifurcations in nonlinear equations, providing a unified framework that extends beyond classical variational systems.

Contribution

It develops an abstract minimax bifurcation formula, proves existence and characterization of bifurcation points, and justifies finite-dimensional approximations, applicable to non-variational systems.

Findings

Established a new minimax bifurcation formula.

Proved existence and characterization of bifurcation points.

Validated the approach with applications to nonlinear elliptic systems.

Abstract

We develop a variational minimax method for detecting maximal saddle-node bifurcations in abstract nonlinear equations. Unlike continuation and path-following techniques, the method identifies the critical parameter directly as an extremal value of an extended Rayleigh quotient. We prove an abstract minimax bifurcation formula, establish the existence and characterization of weak saddle-node bifurcation points, and justify finite-dimensional Galerkin approximations. We also obtain perturbation estimates for the bifurcation value. Applications to non-variational systems of nonlinear elliptic equations show that the approach is not restricted to classical variational structures. The resulting framework provides a unified tool for detecting, approximating, and analyzing saddle-node bifurcations.