Existence of constrained minimizer for a quadratically coupled Schrodinger systems

In this paper, we consider the existence of solutions to a quadratically coupled Schrodinger systems -Delta u = mu(1)u + 2 alpha u v in R-N, -Delta v = mu(2)v + alpha u(2) in R-N under the condition integral(RN) u(2) = a, integral(RN) v(2) = b. Here N = 1, 2, alpha > 0 and a, b> 0 are fixed. In the systems, mu(1) and mu(2) are unknown. We prove that there exists a solution (mu(1), mu(2), (u) over tilde, (v) over tilde) to the systems with mu(1) < 0, mu(2) < 0, and (u) over tilde, (v) over tilde. C-2(R-N) being positive, radially symmetric, and decreasing with r = vertical bar x vertical bar. Our argument is heavily based on the rearrangement techniques.