Abstract: We study the ground states of attractive binary Bose-Einstein condensates with $N$ particles, which are trapped in the steep potential wells $\lambda V(x)$ in $\mathbb{R}^2$. We show that there exists a positive number $N_{*}$ such that if $N>N_{*}$, the system admits no ground state for any $\lambda>0$. Moreover, there exist two positive numbers, $M_{*}$ and $\lambda^*(N)$, such that if $N<M_{*}$, then for any $\lambda>\lambda^*(N)$, the system admits at least one ground state. As $\lambda\rightarrow\infty$, for any fixed $N<M_{*}$, we give a detailed description for the limit behavior of both positive and semi-trivial ground states.

Key words: ground state, binary Bose-Einstein condensate, steep potential well, limit behavior