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

TL;DR

This paper investigates the existence and properties of solutions to coupled Schrödinger equations with critical exponent on waveguide manifolds, extending results to systems with multiple components and different geometries.

Contribution

It introduces new techniques for proving existence and dependence of solutions, generalizing to various manifolds and multi-component systems.

Findings

Existence of solutions with specific y-dependence

Extension to systems with multiple components

Applicability to general 1D Riemannian manifolds

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.