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.

We justify variational principles of a new type corresponding to bifurcations of solutions for families of equations given in variational form. To illustrate the method, we consider elliptic equations with sign-indefinite nonlinearities and prove the existence of pairwise creationannihilation bifurcations of their positive solutions. The corresponding bifurcation points are expressed via explicitly specified variational principles.

We apply the nonlinear generalized Rayleigh quotients method to develop new tools that can be used to study ground states of nonlinear Schrödinger equations. We introduce a new type variational functional, the global minimizer of which corresponds to the so-called fundamental frequency solutions with a prescribed action value. We find the ground state of the problem and uniquely determine the corresponding values of the mass, frequency, and action level. Based on this approach, we obtain new results on the existence and absence of nonnegative solutions to the zero mass problem. Bibliography: 15 titles.

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