Radial Solutions and Phase Separation in a System of Two Coupled Schrödinger Equations

We consider the nonlinear elliptic system ><>: −∆u+ u− u − βvu = 0 in B, −∆v + v − v − βuv = 0 in B, u, v > 0 in B, u = v = 0 on ∂B, where N ≤ 3 and B ⊂ R is the unit ball. We show that, for every β ≤ −1 and k ∈ N, the above problem admits a radially symmetric solution (uβ , vβ) such that uβ − vβ changes sign precisely k times in the radial variable. Furthermore, as β → −∞, after passing to a subsequence, uβ → w and vβ → w− uniformly in B, where w = w −w− has precisely k nodal domains and is a radially symmetric solution of the scalar equation ∆w − w + w = 0 in B, w = 0 on ∂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.