Fundamental Frequency Solutions with Prescribed Action Value to Nonlinear Schrödinger Equations

Abstract

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.

On ground states for the 2D Schrödinger equation with combined nonlinearities and harmonic potential

“…Like in Ref. 20, our main result is in the spirit of the former notion: we prove the orbital stability of the set of fundamental frequency solutions, defined below. Note that by resuming the approach of, for example, Ref.…”

“…Second, in the case without potential, the set of stationary solutions was studied recently in Ref. 20 by using a generalization of the Rayleigh quotient to the nonlinear framework 21 . However, the approach presented in Ref.…”

“…However, the approach presented in Ref. 20 excludes the 2D case (as well as the one‐dimensional case, a situation where many specific techniques are available, see, e.g., Refs. 22, 23).…”

“…The proof of the theorem is based on development of the method of fundamental frequency solutions, introduced by the second author in Ref. 20 in the higher‐dimensional case without potential. We deal with the so‐called prescribed action solution of (), that is, a function $$u^S \in \Sigma$$ which for a given action $$S \in {\mathbb {R}}$$ satisfies $$$$\begin{equation} {S}_{\lambda ,\mu}({u}^{S})=S\ \text{and}\ D{S}_{\lambda ,\mu}({u}^{S})=0, \end{equation}$$$$with some $$\lambda \ge 0$$.…”

“…We obtain the existence of the fundamental frequency solutions by applying the NG‐Rayleigh quotient method20, 21, 44 to () with $$\Lambda ^S_\mu (u)$$ as the Rayleigh quotient. Remark We conjecture that Equation () has no ground state solution with positive action $$S>0$$.…”