Critical surface for two component Bose-Einstein Condensates

Abstract

We investigate the existence of ground states for two-component Bose--Einstein condensates with intraspecies interactions $a_1,a_2\in (0,a^{\ast })$ and interspecies interaction $\beta >0$. By carefully investigating an associated auxiliary minimization problem, we prove the existence of a unique, continuous, critical surface $\gamma =\gamma (a_1,a_2)$ between ${\beta }_{\ast }:=\sqrt{(a^{\ast }-a_1)(a^{\ast }-a_2)}$ and ${\beta }^{\ast }:=a^{\ast }-\frac{a_1+a_2}{2}$. This surface provides a complete classification for the existence of minimizers: ground states exist for $\beta \in (0,\gamma )$ and do not exist for $\beta >\gamma$. Additionally, when the trapping potentials are continuous at a common zero point, there is no ground state for $\beta =\gamma$. Our results close the gap case $\beta \in [{\beta }_{\ast },{\beta }^{\ast }]$ when $a_1\ne a_2$ in literature, significantly extending the results in \cite{Bao-Cai-1,guoBlowupSolutionsTwo2017a}.