Existence and dependency results for coupled Schr\"odinger equations with critical exponent on waveguide manifold

Abstract

We study the coupled Schr\"odinger 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.