On finding bifurcations for nonvariational elliptic systems by the extended quotients method

We develop a new method for finding bifurcations for nonlinear systems of equations based on a direct finding of bifurcations through saddle points of extended quotients. The method is applied to find the saddle-node bifurcation point for system of elliptic equations with the nonlinearity of the general convex-concave type. The main result justifies the variational formula for the detection of the maximal 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.