Semitrivial vs. fully nontrivial ground states in cooperative cubic Schrödinger systems with \(d\geq 3\) equations (Q306563)

Let \(N\in \{1,2,3\}\), \(d\in \mathbb{N}\), and for \(i,j=1,\dots,d\), with \(i\neq j\), let \(\lambda_i,\mu_i>0\) and \(b_{ij}>0\) with \(b_{ij}=b_{ji}\). The authors consider the weakly coupled Schrödinger cubic system NEWLINE\[NEWLINE\begin{aligned} & -\Delta u_i+\lambda_iu_i=\mu_i u_i^3+u_i\sum_{j\neq i}b_{ij}u_j^2, \\ & u_i\in H^1(\mathbb{R}^N,\mathbb{R}), \\ & i=1,\dots,d\end{aligned}NEWLINE\]NEWLINE and study the existence/non existence of fully nontrivial ground state solutions in dependence of the parameters \(\lambda_i,\mu_i, b_{i,j}>0\). A nonzero weak solution to \((1)\) is said fully nontrivial if all of its components are nontrivial, and semitrivial in the opposite case. A weak solution is said ground state if it minimizes the energy functional over the set of all nontrivial weak solutions.NEWLINENEWLINEFor \(b_{ij}=b>0\) and \(d\geq 3\), the authors prove that every ground state solution is fully nontrivial for sufficiently large \(b\) provided that the vector \((\lambda_1,\dots,\lambda_d)\) satisfies one of the two conditions:NEWLINENEWLINE(1) \(\lambda_1\leq \lambda_2\leq\dots\leq \lambda_d\) and NEWLINE\[NEWLINE\max_{2\leq i\leq d}\lambda_i<\alpha \min_{2\leq i\leq d}\lambda_i,NEWLINE\]NEWLINE where \(\alpha\) is a (explicitly computed) number depending on \(\lambda_2/\lambda_1,d,N\);NEWLINENEWLINE(2) \(\max_{1\leq i\leq d}\lambda_i<\frac{d-1}{d-2} \min_{1\leq i\leq d}\lambda_i\).NEWLINENEWLINEFor \(\lambda_1=\lambda_2=\dots=\lambda_d\) and \(d\geq 3\) a similar result is proved under the following assumptions:NEWLINENEWLINE(3) \(\alpha:=\min_{i}(\min_{j}b_{ij}-\mu_i)>0\) and \(\max_{1\leq i \leq d, k\neq j}|b_{ij}-b_{ik}|<\frac{\alpha}{d-2}\).NEWLINENEWLINEAs a counterpart, the cases in which every ground state solution is semitrivial or even trivial are also investigated. In this direction, the authors establish some complementary results to the above ones.