On the Minimax Bifurcation Formula

We introduce a Rayleigh-quotient minimax method for locating maximal one-sided saddle-node bifurcations in nonlinear equations, including non-variational ones. The method avoids branch continuation and instead selects the critical point directly through the minimax bifurcation formula $${\lambda }^{\ast }:=\underset{u\in {\mathcal{U}}^o}{\sup }\underset{v\in \mathcal{S}\setminus \{0\}}{\inf }\mathcal{R}(u,v),$$ where $\mathcal{R}$ is a two-variable extended Rayleigh quotient on fixed cones. A saddle point of this quotient simultaneously determines the critical parameter, the bifurcation solution, and the adjoint singularity relation. This gives a direct characterization of the bifurcation threshold and leads to Galerkin minimax approximations, a posteriori parameter-margin estimates, and perturbation bounds for the critical value. The abstract assumptions are verified for nonlinear elliptic systems, including non-potential systems.