Abstract
In this paper, we study the focusing nonlinear Schr\"odinger equation with exponential nonlinearities \[ i \partial_t u + \Delta u = - \left(e^{4\pi |u|^2} - 1 - 4\pi \mu |u|^2 \right) u, \quad u(0) = u_0 \in H^1, \quad (t,x) \in \mathbb{R} \times \mathbb{R}^2, \] 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 \(t_n\rightarrow +\infty\) and \(R_n \rightarrow \infty\) such that \(u(t_n)\) vanishes inside \(B(0,R_n)\) for all \(n\geq 1\) or the solution scatters in \(H^1\).