Ground states of two-component attractive Bose-Einstein condensates II: Semi-trivial limit behavior

As a continuation of our prior article, we study new pattern formations of ground states $(u_1,u_2)$ for two-component Bose-Einstein condensates (BEC) with homogeneous trapping potentials in $\mathbb {R}^2$, where the intraspecies interaction $(-a,-b)$ and the interspecies interaction $-\beta$ are both attractive, i.e., $a$, $b$, and $\beta$ are all positive. If $0<b<a^*:=\|w\|^2_2$ and $0<\beta <a^*$ are fixed, where $w$ is the unique positive solution of $\Delta w-w+w^3=0$ in $\mathbb {R}^2$, the semi-trivial behavior of $(u_1,u_2)$ as $a\nearrow a^*$ is proved in the sense that $u_1$ concentrates at a unique point and while $u_2\equiv 0$ in $\mathbb {R}^2$. However, if $0<b<a^*$ and $a^*\le \beta <\beta ^*=a^*+\sqrt {(a^*-a)(a^*-b)}$, the refined spike profile and the uniqueness of $(u_1,u_2)$ as $a\nearrow a^*$ 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 $a^*$.

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