On the Cauchy problem of 3-D energy-critical Schrödinger equations with subcritical perturbations

We investigate the global well-posedness, scattering and blow up phenomena when the 3-D quintic nonlinear Schrödinger equation, which is energy-critical, is perturbed by a subcritical nonlinearity λ1upu. We find when the quintic term is defocussing, then the solution is always global no matter what the sign of λ1 is. Scattering will occur either when the perturbation is defocussing and or when the mass of the solution is small enough and . When the quintic term is focusing, we show the blow up for certain solutions.