Semitrivial vs. fully nontrivial ground states in cooperative cubic Schrödinger systems with d ≥ 3 equations

In this work we consider the weakly coupled Schrödinger cubic system{−Δu_i+λ_iu_i=μ_iu_i ³+u_i∑j≠ib_iju_j ²u_i∈H¹(R^N;R),i=1,…,d, where 1≤N≤3, λ_i,μ_i>0 and b_ij=b_ji>0 for i≠j. This system admits semitrivial solutions, that is solutions u=(u₁,…,u_d) with null components. We provide optimal qualitative conditions on the parameters λ_i, μ_i and b_ij under which the ground state solutions have all components nontrivial, or, conversely, are semitrivial. This question had been clarified only in the d=2 equations case. For d≥3 equations, prior to the present paper, only very restrictive results were known, namely when the above system was a small perturbation of the super-symmetrical case λ_i≡λ and b_ij≡b. We treat the general case, uncovering in particular a much more complex and richer structure with respect to the d=2 case.