Uniqueness of positive bound states to Schrodinger systems with critical exponents

We prove the uniqueness for the positive solutions of the following elliptic systems: \begin{eqnarray*} \left\{\begin{array}{ll} - \lap (u(x)) = u(x)^{\alpha}v(x)^{\beta} - \lap (v(x)) = u(x)^{\beta} v(x)^{\alpha} \end{array} \right. \end{eqnarray*} Here $x\in R^n$, $n\geq 3$, and $1\leq \alpha, \beta\leq \frac{n+2}{n-2}$ with $\alpha+\beta=\frac{n+2}{n-2}$. In the special case when $n=3$ and $\alpha =2, \beta=3$, the systems come from the stationary Schrodinger system with critical exponents for Bose-Einstein condensate. As a key step, we prove the radial symmetry of the positive solutions to the elliptic system above with critical exponents.