Existence and bounds for nonlinear Schrödinger systems with strong competition

We consider systems of coupled Schrödinger equations which appear in nonlinear optics and binary Bose–Einstein condensation. Namely, we prove that for most μ,ν∈R, a,b>0, the system −Δu=μu+au³−βuv², −Δv=νv+bv³−βvu², u,v∈H¹₀(B₁(0)), where B₁(0) is the unit ball of R³, admits a family of radially symmetric positive solutions (u_β,v_β) provided the interaction parameter β>0 is sufficiently large. By using a Morse index technique we deduce that these solutions are bounded uniformly in β, hence their limit functions as β→∞ undergo the phenomenon of phase segregation.