Sharp conditions of global existence and scattering for a focusing energy-critical Schrödinger equation

We consider a focusing energy-critical Schrödinger equation with subcritical perturbations and address question related to the sharp criterion of global existence and scattering. By analyzing the variational characteristics of this equation, we established two types of invariant flows. Then approximating this equation by the energy-critical nonlinear Schrödinger equations with the same initial data and combining the properties of the invariant flows, we obtain the sharp conditions of global existence for this equation. Moreover, when the solution is globally defined, we prove the scattering.