  This paper provides a direct method of establishing the existence and
uniqueness of saddle-node bifurcations for nonlinear equations in general
domains. The method employs the scaled extended quotient whose saddle points
correspond to the saddle-node bifurcations. The uniqueness of the saddle-node
bifurcation point directly stems from the uniqueness of the saddle point. The
method is applied to solving open problems involving the existence and
uniqueness of the maximum saddle-node bifurcation for the positive solutions
curve to an elliptic boundary value problem with a convex-concave nonlinearity
in general domains.
