Existence and uniqueness of a saddle-node bifurcation point for nonlinear equations in general domains

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.