Asymptotic properties of non-relativistic limit for pseudo-relativistic Hartree equations

In this paper, we study the asymptotic behavior of energy and action ground states to the following pseudo-relativistic Hartree equation $$\left(\sqrt{-c^2\Delta +m^2c^4}-mc^2\right)u + \lambda u = \left(|x|^{-1}*|u|^2\right)u$$ as the speed of light $c\to\infty$. We obtain an asymptotic expansion of the ground state as $c \to \infty,$ which is new in the case of the energy ground state and generalizes the results of Choi, Hong, and Seok (2018) for the action ground state.