Ground States of Attractive Fermi Schr\"{o}dinger Systems with Ring-Shaped Potentials

Abstract

As an application of the finite-rank Lieb-Thirring inequality established in [R. L. Frank, D. Gontier and M. Lewin, Comm. Math. Phys., 2021], we study ground states of mass-critical N-coupled Fermi nonlinear Schr\"{o}dinger systems with attractive interactions in $\mathbb{R}^3$, which are trapped in ring-shaped potentials. For any given $N\in\mathbb{N}^+$, we prove that ground states exist if $0<a<a_N^*$, where $a$ denotes the strength of attractive interactions in the system, and $a_N^*$ is the best constant of a finite-rank Lieb-Thirring inequality. Moreover, for some $N\in\mathbb{N}^+$, we also prove the nonexistence of minimizers for the system as soon as $a\geq a_N^*$. Applying the energy estimates and the blow-up analysis, we further analyze the mass concentration behavior of ground states for the system as $a\nearrow a_N^*$.