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 \[ λ^* := \sup_{u\in\mathcal U^o} \inf_{v\in\mathcal S\setminus\{0\}} \mathcal R(u,v), \] 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.