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

Abstract

In this paper, we consider the following nonlinear Schrödinger system: -$\hat{I}”$ u+P(x)u=$\hat{I}{¼}_1$ $u^3$+${\hat{I}}^2$ u$v^2$, x $\in$ $R^3$,\\ -$\hat{I}”$ v+Q(x)v=$\hat{I}{¼}_2$ $v^3$+${\hat{I}}^2$ $u^2$v, x $\in$ $R^3$, where $P(x),Q(x)$ are positive radial potentials,~$\hat{I}{¼}_1,\hat{I}{¼}_2>0$, ${\hat{I}}^2$ $\in$ $R$ 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.