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

TL;DR

This paper investigates the asymptotic behavior of ground states in pseudo-relativistic Hartree equations as the speed of light approaches infinity, providing new expansions and generalizing previous results.

Contribution

It introduces a novel asymptotic expansion for the energy ground state and extends prior work on action ground states in the non-relativistic limit.

Findings

Derived a new asymptotic expansion for energy ground states

Generalized previous results for action ground states

Enhanced understanding of non-relativistic limits in Hartree equations

Abstract

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.