Radial solutions and phase separation in a system of two coupled Schrodinger equations

We consider the nonlinear elliptic system [GRAPHICS] where N <= 3 and B subset of R(N) is the unit ball. We show that, for every beta <= -1 and k epsilon N, the above problem admits a radially symmetric solution (u(beta), upsilon(beta)) such that u(beta) - upsilon(beta) changes sign precisely k times in the radial variable. Furthermore, as beta -> -infinity, after passing to a subsequence, u(beta) -> omega(+) and upsilon(beta) -> omega(-) uniformly in B, where omega = omega(+) - omega(-) has precisely k nodal domains and is a radially symmetric solution of the scalar equation Delta omega - omega + omega(3) = 0 in B, omega = 0 on partial derivative B. Within a Hartree-Fock approximation, the result provides a theoretical indication of phase separation into many nodal domains for Bose-Einstein double condensates with strong repulsion.