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In this article, we study the multiplicity, asymptotics and stability ofstanding waves with prescribed mass $c>0$ for nonlinear Schr\"odinger equationwith rotation in the mass-supercritical regime arising in Bose-Einsteincondensation. Under suitable restriction on the rotation frequency, bysearching critical points of the corresponding energy functional on themass-sphere, we obtain a local minimizer $u_c$ and a mountain pass solution$\hat{u}_c$. %under suitable assumptions on the related parameters.Furthermore, we show that $u_c$ is a ground state for small mass $c>0$ anddescribe a mass collapse behavior of the minimizers as $c\to 0$, while$\hat{u}_c$ is an excited state. Finally, we prove that the standing waveassociated with $u_c$ is stable. Notice that the pioneering works\cite{aMsC,shYZ} imply that finite time blow-up of solutions to this modeloccurred in the mass-supercritical setting, therefore, we in the present paperobtain a new stability result. The main contribution of this paper is to extendthe main results in \cite{JeSp,gYlW} concerning the same model frommass-subcritical and mass-critical regimes to mass-supercritical regime, wherethe physically most relevant case is covered.

Multiplicity, asymptotics and stability of standing waves for nonlinear Schr\"odinger equation with rotation