On the Minimax Bifurcation Formula

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.