Ground States of Two-Component Attractive Bose-Einstein Condensates II: Semi-trivial Limit Behavior

As a continuation of [14], we study new pattern formations of ground states $(u_1,u_2)$ for two-component Bose-Einstein condensates (BEC) with homogeneous trapping potentials in $R^2$, where the intraspecies interaction $(-a,-b)$ and the interspecies interaction $-\beta$ are both attractive, $i\mathrm{.}e,$ $a$, $b$ and $\beta$ are all positive. If $0<b<a^{\ast }:=\mathrm{\parallel }w{\mathrm{\parallel }}_2^2$ and $0<\beta <a^{\ast }$ are fixed, where $w$ is the unique positive solution of $\mathrm{\Delta }w-w+w^3=0$ in $R^2$, the semi-trivial behavior of $(u_1,u_2)$ as $a\nearrow a^{\ast }$ is proved in the sense that $u_1$ concentrates at a unique point and while $u_2\equiv 0$ in $R^2$. However, if $0<b<a^{\ast }$ and $a^{\ast }\le \beta <{\beta }^{\ast }=a^{\ast }+\sqrt{(a^{\ast }-a)(a^{\ast }-b)}$, the refined spike profile and the uniqueness of $(u_1,u_2)$ as $a\nearrow a^{\ast }$ are analyzed, where $(u_1,u_2)$ must be unique, $u_1$ concentrates at a unique point, and meanwhile $u_2$ can either blow up or vanish, depending on how $\beta$ approaches to $a^{\ast }$.