Existence and non-existence of normalized solutions for a nonlinear fractional Schrödinger system

This paper is concerned with a nonlinear fractional Schördinger system in $ \mathbb{{R}} $ with intraspecies interactions $ a_{i}>0 \ (i = 1,2) $ and interspecies interactions $ \beta \in\mathbb{{R}} $. We study this system by solving an associated constrained minimization problem (i.e., $ L^2- $norm constaints). Under certain assumptions on the trapping potentials $ V_i(x) \ (i = 1,2), $ we derive some delicate estimates for the related energy functional and establish a criterion for the existence and non-existence of solutions, in which way several existence results are obtained.