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 λ₁p|u|u. 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.