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 $$(\sqrt{-c^2\mathrm{\Delta }+m^2{c}^4}-m{c}^2)u+\lambda u=(\mathrm{\mid }x{\mathrm{\mid }}^{-1}\ast \mathrm{\mid }u{\mathrm{\mid }}^2)u$$ as the speed of light $c\to \mathrm{\infty }$. We obtain an asymptotic expansion of the ground state as $c\to \mathrm{\infty },$ 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.