Ground States of Two-Component Attractive Bose-Einstein Condensates I: Existence and Uniqueness

Abstract

We study ground states of two-component Bose-Einstein condensates (BEC) with trapping potentials in $R^2$, where the intraspecies interaction $(-a_1,-a_2)$ and the interspecies interaction $-\beta$ are both attractive, $i.e,$ $a_1$, $a_2$ and $\beta$ are all positive. The existence and non-existence of ground states are classified completely by investigating equivalently the associated $L^2$-critical constraint variational problem. The uniqueness and symmetry-breaking of ground states are also analyzed under different types of trapping potentials as $\beta \nearrow {\beta }^{\ast }=a^{\ast }+\sqrt{(a^{\ast }-a_1)(a^{\ast }-a_2)}$, where $0<a_i<a^{\ast }:=\Vert w{\Vert }_2^2$ ($i=1,2$) is fixed and $w$ is the unique positive solution of $\Delta w-w+w^3=0$ in $R^2$. The semi-trivial limit behavior of ground states is tackled in the companion paper [12].

40 pages, two figures