Global well-posedness, scattering and blow-up for the energy-critical, Schrödinger 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, Schrödinger equation with general nonlinearity \begin{equation*} \left{\begin{array}{l} i \partial_t u+Δu + f(u)=0,\ (x, t) \in $\mathbb{R}$^N \times $\mathbb{R}$, \ \left.u\right|_{t=0}=u_0 \in H ^1($\mathbb{R}$^N), \end{array}\right. \end{equation*} where $f:\mathbb{C}\to \mathbb{C}$ 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 \cite{KCEMF2006}(Invent. Math).