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

Abstract

In this work we consider the weakly coupled Schrödinger cubic system [vg. equation in attachment] where 1 ≤ N ≤ 3, λi, μi > 0 and bij = bji > 0 for i ≠ j. This system admits semitrivial solutions, that is solutions u = (u1, ..., ud) with null components. We provide optimal qualitative conditions on the parameters λi, μi and bij 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 λi ≡ λ and bij ≡ b. We treat the general case, uncovering in particular a much more complex and richer structure with respect to the d = 2 case