On the Limiting Behavior of $L^2$-Critical Pseudo-Relativistic Fermi Systems

We consider ground states of a pseudo-relativistic Fermi system in the $L^2$-critical case. We prove that the system admits ground states, if and only if the attractive strength $a$ satisfies $0<a<D_{4/3,2}$, where $D_{4/3,2}\in(0, \infty)$ is the optimal constant of a dual fractional Lieb--Thirring inequality. The limiting behavior of ground states for the system is further analyzed as $a\nearrow D_{4/3,2}$. As a byproduct, the qualitative properties of optimizers for the dual fractional Lieb-Thirring inequality are also investigated.