Global well-posedness, scattering and blow-up for the energy-critical, Schr\"odinger equation with general nonlinearity in the radial case

In this paper, we study the well-posedness theory and the scattering asymptotics for the energy-critical, Schr\"odinger equation with general nonlinearity equation* \array{l i ₜ u+ u + f (u) =0, \ (x, t) RN R, \\. u|ₓ=₀=u₀ H ¹ (RN), array. equation* where f: C C satisfies Sobolev critical growth condition. Using contraction mapping method and concentration compactness argument, we obtain the well-posedness theory in proper function spaces and scattering asymptotics. This paper generalizes the conclusions in KCEMF2006 (Invent. Math).