  We study the qualitative properties of a limiting elliptic system arising in
phase separation for Bose-Einstein condensates with multiple states: \Delta u=u
v^2 in R^n, \Delta v= v u^2 in R^n, u, v>0\quad in R^n. When n=1, we prove
uniqueness of the one-dimensional profile. In dimension 2, we prove that stable
solutions with linear growth must be one-dimensional. Then we construct entire
solutions in $\R^2$ with polynomial growth $|x|^d$ for any positive integer $d
\geq 1$. For $d\geq 2$, these solutions are not one-dimensional. The
construction is also extended to multi-component elliptic systems.
