Positive and sign-changing solutions for the nonlinear Schrödinger systems with synchronization and separation

In this paper, we consider the following nonlinear Schrödinger system: -$\mathrm{\Delta }$ u+P(x)u=${\mu }_1$ $u^3$+$\beta$ u$v^2$, x $\in$ $R^3$,\ -$\mathrm{\Delta }$ v+Q(x)v=${\mu }_2$ $v^3$+$\beta$ $u^2$v, x $\in$ $R^3$, where $P(x),Q(x)$ are positive radial potentials,~${\mu }_1,{\mu }_2>0$, $\beta$ $\in$ $R$ is a coupling constant. We constructed a new type of solutions which are different from the ones obtained in \cite{PW}. This new family of solutions to system have a more complex concentration structure and are centered at the points lying on the top and the bottom circles of a cylinder with height $h$. Moreover, we examine the effect of nonlinear coupling on the solution structure. In the repulsive case, we construct an unbounded sequence of non-radial positive vector solutions of segregated type. In the attractive case, we construct an unbounded sequence of non-radial positive vector solutions of synchronized type. Moreover, we prove that there exist infinitely many sign-changing solutions whose energy can be arbitrarily large.