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

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 $a^{\ast }$. %attractive intraspecies interactions satisfies $a_i<$, where $Q$ is the unique positive radial solution of $\mathrm{\Delta }u-u+u^3=0$ in ${\mathbb{R}}^2$; in contrast, there is no minimizer if either $a_i>a^{\ast }$ for $i=1$ or $2$, or $a_1=a_2=a^{\ast }$. Furthermore, as $(a_1,a_2)\nearrow (a^{\ast },a^{\ast })$, 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.