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

Abstract

In this paper, we study the following two-component systems of nonlinear Schrödinger equations \begin{equation*} \left\{\aligned&Δu-(λa(x)+a_0(x))u+μ_1u^3+βv^2u=0\quad&\text{in }\bbr^3,\\ &Δv-(λb(x)+b_0(x))v+μ_2v^3+βu^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 $\lambda \to +\infty$ and $\beta \to -\infty$ are also considered. In particular, the phenomenon of phase separations is observed in the whole space $\bbr^3$. In the Hartree-Fock theory, this provides a theoretical enlightenment of phase separation in $\bbr^3$ for the 2-mixtures of Bose-Einstein condensates.

39 pages