The monotonicity conjecture and stability of solitons for the Cubic-Quintic NLS on R^3

In this paper, we prove stability or instability of solitons for the cubic-quintic nonlinear Schrodinger equation at every frequency. The monotonicity conjecture raised by Killip, Oh, Pocovnicu and Visan is resolved. We introduce and solve a new cross-constrained variational problem. Then the uniqueness of the energy minimizers is proposed and shown. According to a spectral approach and variational arguments, we develop a set of geometric analysis methods. Correspondences between the soliton frequency and the prescribed mass are established. Classification of normalized solutions is given.