Existence and dependency results for coupled Schrödinger equations with critical exponent on waveguide manifold

We study the coupled Schrödinger equations with critical exponent on ${\mathbb{R}}^3\times \mathbb{T}$. With the help of scaling argument and semivirial-vanishing technology, we obtain the existence and $y$-dependence of solution, the tori can be generalized to $1$-dimensional compact Riemannian manifold. Moreover, the conclusion of this paper can be extended to systems with any number of components.