  Using the concentration-compactness method and the localized virial type
arguments, we study the behavior of $H^1$ solutions to the focusing quintic NLS
in $\R^2$, namely, $$i \partial_t u+\Delta u+|u|^4u=0,\quad\quad (x, t) \in
\R^2\times\R.$$
  Denoting by $M[u]$ and $E[u]$, the mass and energy of a solution $u,$
respectively, and $Q$ the ground state solution to $-Q+\Delta Q+ |Q|^4Q=0$, and
assuming $M[u]E[u] <M[Q]E[Q]$, we characterize the threshold for global versus
finite time existence. Moreover, we show scattering for global existing time
solutions and finite or "weak" blow up for the complement region. This work is
in the spirit of Kenig and Merle and Duyckaerts, Holmer, and Roudenko.
