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

In this paper, we consider the following nonlinear Schr & ouml;dinger system: {Delta u+P(x)u=mu(1)u(3)+beta uv(2),x is an element of R-3, {Delta v+Q(x)v=mu(2)v(3)+beta u(2)v,x is an element of R-3, where P(x),Q(x) are positive radial potentials, mu(1),mu(2)>0, beta is an element of R is a coupling constant. We constructed a new type of solutions which are different from the ones obtained by Peng and Wang [Arch Rational Mech Anal. 2013;208:305-339]. 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. Meanwhile, 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.