We consider the nonlinear Schrödinger equation (NLSH) with the convolution combined term
(CNLSH) in the energy space
. We firstly use a variational approach to give a dichotomy of scattering and blow up for the radial solution with the energy below the threshold, which is given by the ground state W for the energy‐critical NLS: iut+Δu=−|u|4u. The basic strategy is the concentration‐compactness arguments from Kenig and Merle. We overcome the main difficulties coming from the lack of scaling invariance and the non‐local property of the convolution term. Our result shows that the focusing,
‐critical term −|u|4u plays the decisive role of the threshold of the scattering solution of (CNLSH) in the energy space. Copyright © 2015 John Wiley & Sons, Ltd.