  We develop a novel method for finding bifurcations for nonlinear systems of
equations based on directly finding bifurcations through saddle points of
extended quotients. The method is applied to find the saddle-node bifurcation
point for elliptic equations with the nonlinearity of the general
convex-concave type. The main result justifies the variational formula for the
detection of the maximum saddle-node type bifurcation point of stable positive
solutions. As a consequence, a precise threshold value separating the interval
of the existence of stable positive solutions is established.
