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We consider a nonlinear Schr\"odinger system arising in a two-componentBose-Einstein condensate (BEC) with attractive intraspecies interactions andrepulsive interspecies interactions in $\mathbb{R}^2$. We get ground states ofthis system by solving a constrained minimization problem. For some kinds oftrapping potentials, we prove that the minimization problem has a minimizer ifand only if the attractive interaction strength $a_i (i=1,2)$ of each componentof the BEC system is strictly less than a threshold $a^*$. %attractiveintraspecies 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 $(a_1, a_2)\nearrow (a^*, a^*)$, theasymptotical behavior for the minimizers of the minimization problem isdiscussed. Our results show that each component of the BEC system concentratesat a global minimum of the associated trapping potential.

Blow-up behavior of ground states for a nonlinear Schr\"{o}dinger system with attractive and repulsive interactions