Dynamics of subcritical threshold solutions for energy-critical NLS

In this paper, we study the dynamics of subcritical threshold solutions for focusing energy critical NLS on ${\mathbb{R}}^d$ ($d\ge 5$) with nonradial data. This problem with radial assumption was studied by T. Duyckaerts and F. Merle in \cite{DM} for $d=3,4,5$ and later by D. Li and X. Zhang in \cite{LZ} for $d \geq 6$. We generalize the conclusion for the subcritical threshold solutions by removing the radial assumption for $d\ge 5$. A key step is to show exponential convergence to the ground state $W(x)$ up to symmetries if the scattering phenomenon does not occur. Remarkably, an interaction Morawetz-type estimate are applied.