UNIQUENESS OF POSITIVE BOUND STATES TO SCHRODINGER SYSTEMS WITH CRITICAL EXPONENTS

We prove the uniqueness of the positive solutions of the following elliptic system: (1) -Delta(u(x)) = u(x)(alpha)v(x)(beta), (2) -Delta(v(x)) = u(x)(beta)v(x)(alpha). Here x epsilon R-n, n >= 3, and 1 <= alpha < beta <= n+2/n-2 with alpha + beta = n+2/n-2. In the special case when n = 3 and alpha = 2, beta = 3, the system is closely related to the ones from the stationary Schrodinger system with critical exponents for the Bose-Einstein condensate. As the first step, we prove the radial symmetry of the positive solutions to the elliptic system above with critical exponents. We then prove that u = v, which is a key point for our uniqueness result.