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

In this paper, we investigate bound states with prescribed angular momentum and mass for the nonlinear Schrödinger 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.