In this paper, we investigate the well-posedness and ill-posedness issues for the incompressible stationary Hall-magnetohydrodynamic (Hall-MHD) system in \(\mathbb {R}^3.\) We first show the existence and uniqueness of solutions provided with the forces in \(\dot{B}^{3/p-3}_{p,r}(\mathbb {R}^3)\) for \(1\le p <3\) and \(r=1\) . Moreover, this result can be extended to any \(1\le r\le \infty \) whenever \(p=2,\) without any additional assumption on the physical parameters. On the other hand, we establish some ill-posedness results for Hall-MHD system by using the discontinuity of the solution mapping of the three-dimensional stationary Navier–Stokes equations in critical function spaces \(\dot{B}^{3/p-1}_{p,r}(\mathbb {R}^3)\) ( \(p\ge 3\) ).
