Long time dynamics for the focusing nonlinear Schr\"odinger equation with exponential nonlinearities

In this paper, we study the focusing nonlinear Schr"odinger equation with exponential nonlinearities $$i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u=-(e^{4\pi \mathrm{\mid }u{\mathrm{\mid }}^2}-1-4\pi \mu \mathrm{\mid }u{\mathrm{\mid }}^2)u,u(0)=u_0\in H^1,(t,x)\in \mathbb{R}\times {\mathbb{R}}^2,$$ where $\mu \in \{0,1\}$. By using variational arguments, we first derive invariant sets where the global existence and finite time blow-up occur. In particular, we obtain sharp thresholds for global existence and finite time blow-up. In the case $\mu =1$, by adapting a recent argument of Arora-Dodson-Murphy \cite{ADM}, we study the long time dynamics of global solutions. It turns out that either there exist $t_n\to +\mathrm{\infty }$ and $R_n\to \mathrm{\infty }$ such that $u(t_n)$ vanishes inside $B(0,R_n)$ for all $n\ge 1$ or the solution scatters in $H^1$.