  We consider a nonlinear Schr\"odinger system arising in a two-component
Bose-Einstein condensate (BEC) with attractive intraspecies interactions and
repulsive interspecies interactions in $\mathbb{R}^2$. We get ground states of
this system by solving a constrained minimization problem. For some kinds of
trapping potentials, we prove that the minimization problem has a minimizer if
and only if the attractive interaction strength $a_i (i=1,2)$ of each component
of the BEC system is strictly less than a threshold $a^*$. %attractive
intraspecies interactions satisfies $a_i< %a^*= \|Q\|_2^2,\ i=1,\,2$, where $Q$
is the unique positive radial solution of $\Delta u-u+u^3=0$ in $\mathbb{R}^2$;
in contrast, there is no minimizer if either $a_i > a^*$ for $i=1$ or $2$, or
$a_1=a_2=a^*$. Furthermore, as $(a_1, a_2)\nearrow (a^*, a^*)$, the
asymptotical behavior for the minimizers of the minimization problem is
discussed. Our results show that each component of the BEC system concentrates
at a global minimum of the associated trapping potential.
