Ground States of Fermionic Nonlinear Schrödinger Systems with Coulomb Potential II: The $L^2$-Critical Case

As a continuation of \cite{me}, we consider ground states of the $N$ coupled fermionic nonlinear Schrödinger system with a parameter $a$ and the Coulomb potential $V(x)$ in the $L^2$-critical case, where $a>0$ represents the attractive strength of the quantum particles. For any given $N\in {\mathbb{N}}^{+}$, we prove that the system admits ground states, if and only if the attractive strength $a$ satisfies $0<a<a_N^{\ast }$, where the critical constant $0<a_N^{\ast }<\mathrm{\infty }$ is the same as the best constant of a dual finite-rank Lieb-Thirring inequality. By developing the so-called blow-up analysis of many-body fermionic problems, we also prove the mass concentration behavior of ground states for the system as $a\nearrow a_N^{\ast }$.