On a two-component Bose-Einstein condensate with steep potential wells

In this paper, we study the following two-component systems of nonlinear Schr"odinger equations \begin{equation*} \left{\aligned&\Delta u-(\lambda a(x)+a_0(x))u+\mu_1u^3+\beta v^2u=0\quad&$\text{in }$\bbr^3,\ &\Delta v-(\lambda b(x)+b_0(x))v+\mu_2v^3+\beta u^2v=0\quad&$\text{in }$\bbr^3,\ &u,v\in\h,\quad u,v>0\quad$\text{in }$\bbr^3,\endaligned\right. \end{equation*} where $\lambda ,{\mu }_1,{\mu }_2>0$ and $\beta <0$ are parameters; $a(x),b(x)\ge 0$ are steep potentials and $a_0(x),b_0(x)$ are sign-changing weight functions; $a(x)$, $b(x)$, $a_0(x)$ and $b_0(x)$ are not necessarily to be radial symmetric. By the variational method, we obtain a ground state solution and multi-bump solutions for such systems with $\lambda$ sufficiently large. The concentration behaviors of solutions as both $\lambda \to +\mathrm{\infty }$ and $\beta \to -\mathrm{\infty }$ are also considered. In particular, the phenomenon of phase separations is observed in the whole space ${\text{bbr}}^3$. In the Hartree-Fock theory, this provides a theoretical enlightenment of phase separation in ${\text{bbr}}^3$ for the 2-mixtures of Bose-Einstein condensates.