Axial Symmetry and Refined Spike Profiles of Ground States for Rotating Two-Component Bose Gases

Abstract

This paper considers ground states of two-component Bose gases confined in an anharmonic trap rotating at the velocity $0<\Omega <\infty$, where the intraspecies interaction $(-a_1,-a_2)$ and the interspecies interaction $-\beta$ are both attractive, i.e, $a_1,a_2$ and $\beta$ are all positive. We prove the axially symmetry and the refined spike profiles of ground states as $\beta \nearrow {\beta }^{\ast }:=a^{\ast }+\sqrt{(a^{\ast }-a_1)(a^{\ast }-a_2)}$, where $0<a_1,a_2<a^{\ast }:=\Vert Q{\Vert }_2^2$ are fixed and $Q>0$ is the unique positive solution of $-\Delta u+u-u^3=0$ in ${\mathbb{R}}^2$.