Infinitely many solutions with simultaneous synchronized and segregated components for nonlinear Schrödinger systems

In this paper, we consider the following nonlinear Schrödinger system in R³: −Δu_j+P_j(x)u=μ_ju_j³+∑i=1,i≠jNβ_iju_i²u_j, where N≥3, P_j are nonnegative radial potentials, μ_j>0, and β_ij=β_ji are coupling constants. This type of systems has been widely studied in the last decade, many purely synchronized or segregated solutions are constructed, but few considerations for simultaneous synchronized and segregated solutions exist. On the other hand, there are new challenges in dealing with the existence of multiple sign-changing solutions or semi-nodal solutions. Using Lyapunov-Schmidt reduction method, we construct new type of positive and sign-changing solutions with simultaneous synchronization and segregation. Comparing to known results in the literature, the novelties are multi-fold. 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 general number of components, i.e. for any N≥3.