Infinitely many solutions with simultaneous synchronized and segregated components for nonlinear Schr\"{o}dinger systems

In this paper, we consider the following nonlinear Schr\"odinger system in $R^3$: \begin{align*} -\Delta u_j +P_j(x) u=\mu_j u_j^3+\sum\limits_{i=1,i\neq j}^N\beta_{ij}u_i^2u_j, \end{align*} where $N\geq3$, $P_j$ are nonnegative radial potentials, $\mu_j>0$ and $\beta_{ij}=\beta_{ji}$ are coupling constants. This type of systems have been widely studied in the last decade, many purely synchronized or segregated solutions are constructed, but few considerations for simultaneous synchronized and segregated positive solutions exist. Using Lyapunov-Schmidt reduction method, we construct new type of solutions with simultaneous synchronization and segregation. Comparing to known results in the literature, the novelties are threefold. We prove the existence of infinitely many non-radial positive and also sign-changing vector solutions, where some components are synchronized but segregated with other components; the energy level can be arbitrarily large; and our approach works for any $N \geq 3$.