Positive and sign-changing solutions for the nonlinear Schr\"{o}dinger systems with synchronization and separation

In this paper, we consider the following nonlinear Schr\"odinger system: -Δ u+P(x)u=μ₁ u³+β uv², x ∈ R³,\\ -Δ v+Q(x)v=μ₂ v³+β u²v, x ∈ R³, where $P(x),Q(x)$ are positive radial potentials,~μ₁,\,μ₂>0, β ∈ R is a coupling constant. We constructed a new type of solutions which are different from the ones obtained in {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.