Semitrivial vs. fully nontrivial ground states in cooperative cubic Schrödinger systems with $d\ge3$ equations

Abstract

In this work we consider the weakly coupled Schrödinger cubic system \[ \begin{cases} \displaystyle -Δu_i+λ_i u_i= μ_i u_i^{3}+ u_i\sum_{j\neq i}b_{ij} u_j^2 \\ u_i\in H^1(\mathbb{R}^N;\mathbb{R}), \quad i=1,\ldots, d, \end{cases} \] where $1\leq N\leq 3$, $λ_i,μ_i >0$ and $b_{ij}=b_{ji}>0$ for $i\neq j$. This system admits semitrivial solutions, that is solutions $\mathbf{u}=(u_1,\ldots, u_d)$ 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 $d\geq 3$ 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 $λ_i\equiv λ$ and $b_{ij}\equiv b$. We treat the general case, uncovering in particular a much more complex and richer structure with respect to the $d=2$ case.

23 pages