Fundamental Frequency Solutions with Prescribed Action Value to Nonlinear Schrödinger Equations

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.