We find a normalized solution \(u=(u_1,\ldots ,u_K)\) to the system of K coupled nonlinear Schrödinger equations \(\begin{aligned} \left\{ \begin{array}{l} -\Delta u_i+ \lambda _i u_i = \sum _{j=1}^K\beta _{i,j}u_i|u_i|^{p/2-2}|u_j|^{p/2} \quad \textrm{in} \ \mathbb {R}^3,\\ u_i \in H^1_{rad}(\mathbb {R}^3),\\ \int _{\mathbb {R}^3} |u_i|^2 \, dx = \rho _i^2 \quad \text {for }i=1,\ldots , K, \end{array} \right. \end{aligned}\) where \(\rho =(\rho _1,\ldots ,\rho _K)\in (0,\infty )^K\) is prescribed, \((\lambda ,u) \in \mathbb {R}^K\times H^1(\mathbb {R}^3)^K\) are the unknown and \(4\le p<6\) . In the case of two equations we show the existence of multiple solutions provided that the coupling is sufficiently large. We also show that for negative coupling there are no ground state solutions. The main novelty in our approach is that we use the Cwikel-Lieb-Rozenblum theorem in order to estimate the Morse index of a solution as well as a Liouville-type result in an exterior domain.
