The dynamics of the NLS with the combined terms in five and higher dimensions

In this paper, we continue the study in \cite{MiaoWZ:NLS:3d Combined} to show the scattering and blow-up result of the solution for the nonlinear Schr"{o}dinger equation with the energy below the threshold $m$ in the energy space $H^1({\mathbb{R}}^d)$, iu_t + \Delta u = -|u|^{4/(d-2)}u + |u|^{4/(d-1)}u, ; d\geq 5. \tag{CNLS} The threshold is given by the ground state $W$ for the energy-critical NLS: $i{u}_t+\mathrm{\Delta }u=-\mathrm{\mid }u{\mathrm{\mid }}^{4\mathrm{/}(d-2)}u$. Compared with the argument in \cite{MiaoWZ:NLS:3d Combined}, the new ingredient is that we use the double duhamel formula in \cite{Kiv:Clay Lecture, TaoVZ:NLS:mass compact} to lower the regularity of the critical element in $L_t^{\mathrm{\infty }}{H}_x^1$ to $L^{\mathrm{\infty }}{\dot{H}}_x^{-ϵ}$ for some $ϵ>0$ in five and higher dimensions and obtain the compactness of the critical element in $L_x^2$, which is used to control the spatial center function $x(t)$ of the critical element and furthermore used to defeat the critical element in the reductive argument.