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We consider the focusing mass supercritical nonlinear Schr\"odinger equationwith 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 quantummechanical angular momentum operator. We establish conditions for globalexistence and blow-up in the energy space. Moreover, we prove stronginstability of standing waves under certain conditions on the rotation and thefrequency of the wave. Finally, we construct orbitally stable standing wavessolutions by considering a suitable local minimization problem. Those resultsare obtained for nonlinearities which are $L^{2}$-supercritical.

Global well-posedness, blow-up and stability of standing waves for supercritical NLS with rotation