Three bound states with prescribed angular momentum to the cubic-quintic NLS equations in $\mathbb{R}^{3}$

TL;DR

This paper establishes the existence of three distinct bound states with prescribed angular momentum for the cubic-quintic nonlinear Schrödinger equation in three dimensions, using variational methods and minimax techniques.

Contribution

It introduces a novel approach to find three solutions to the constrained NLS problem, including a new mountain pass path and energy level comparison.

Findings

Existence of three solutions: local minimizer, mountain pass, and global minimizer.

Construction of a new mountain pass path via minimax method.

First demonstration of three solutions for this class of NLS equations.

Abstract

In this paper, we investigate bound states with prescribed angular momentum and mass for the nonlinear Schr\"{o}dinger equations (NLS) with the cubic-quintic nonlinearity in dimensions three. We demonstrate that there exist three solutions for the double constrained problem: a local minimizer, a mountain pass type solution, and a global minimizer. Moreover, by means of the minimax method, we construct a new mountain pass path and further obtain the geometric link among the three solutions as well as a comparison of their energy levels. This seems to be the first paper concerning three solutions, with the method also being applicable to the single constraint problem.