Infinitely many nonradial positive solutions for two-coupled nonlinear Schrödinger equations in N without symmetry

In this paper, we consider the following two-coupled nonlinear Schrödinger equations: {equation*} {cases} &-Δu+P(x)u=μu^3+βuv^2, ^N,\\ &-Δv+Q(x)v=νv^3+βu^2v, ^N,\\ &u,\,v∈ H^1(^N) {cases} {equation*} where $N=2,3$, $P(x)$ and $Q(x)$ are potentials, $μ$ and $ν$ are positive coefficients and β∈\0\ is the coupling parameter. By developing the intermediate reduction arguments in [del Pino-Wei-Yao, Calc. Var., 2015], we prove the existence of infinitely many nonradial positive solutions to the above system under suitable regularity and decay conditions on $P(x)$ and $Q(x)$, { but without any symmetry assumptions on them}. The scalar version of our result also gives { a positive answer} to the first part of [Question~3, del Pino-Wei-Yao, Calc. Var., 2015] and our method might be helpful in studying other problems.