  We consider the focusing mass supercritical nonlinear Schr\"odinger equation
with rotation \begin{equation*}
  iu_{t}=-\frac{1}{2}\Delta u+\frac{1}{2}V(x)u-|u|^{p-1}u+L_{\Omega}u,\quad
(x,t)\in \mathbb{R}^{N}\times\mathbb{R}, \end{equation*} where $N=2$ or $3$ and
$V(x)$ is an anisotropic harmonic potential. Here $L_{\Omega}$ is the quantum
mechanical angular momentum operator. We establish conditions for global
existence and blow-up in the energy space. Moreover, we prove strong
instability of standing waves under certain conditions on the rotation and the
frequency of the wave. Finally, we construct orbitally stable standing waves
solutions by considering a suitable local minimization problem. Those results
are obtained for nonlinearities which are $L^{2}$-supercritical.
