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We study the qualitative properties of a limiting elliptic system arising inphase separation for Bose-Einstein condensates with multiple states: \Delta u=uv^2 in R^n, \Delta v= v u^2 in R^n, u, v>0\quad in R^n. When n=1, we proveuniqueness of the one-dimensional profile. In dimension 2, we prove that stablesolutions with linear growth must be one-dimensional. Then we construct entiresolutions 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. Theconstruction is also extended to multi-component elliptic systems.

On Entire Solutions of an Elliptic System Modeling Phase Separations