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

In this work we consider the weakly coupled Schrodinger cubic system {-Delta u(i) + lambda(i)u(i) = mu(i)u(i)(3) + u(i) Sigma(j not equal i) b(ij)u(j)(2) u(i) is an element of H-1(R-N ; R), i = 1,...,d, where 1 <= N <= 3, lambda(i), mu(i) > 0 and b(ij) = b(ij) > 0 for i not equal j. This system admits semitrivial solutions, that is solutions u = (u(1) , . . . ,u(d)) with null components. We provide optimal qualitative conditions on the parameters lambda(i), mu 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 lambda(i) equivalent to lambda and b(ij) equivalent to b. We treat the general case, uncovering in particular a much more complex and richer structure with respect to the d = 2 case. (C) 2016 Elsevier Inc. All rights reserved.