Ground states of a coupled pseudo-relativistic Hartree system: existence and concentration behavior

TL;DR

This paper studies the existence, non-existence, and concentration behavior of ground states in a coupled pseudo-relativistic Hartree system with trapping potentials, analyzing how solutions behave as coupling strength varies.

Contribution

It provides a complete classification of ground state existence and non-existence, and details the concentration and blow-up behavior of minimizers under certain potential conditions.

Findings

Complete classification of ground states existence

Analysis of blow-up and concentration behavior

Identification of optimal blow-up rates

Abstract

This paper is concerned with the ground states of a coupled pseudo-relativistic Hartree system in $\mathbb{R} ^{3} $ with trapping potentials, where the intraspecies and the interspecies interaction are both attractive. By investigating an associated constraint minimization problem, the existence and non-existence of ground states are classified completely. Under certain conditions on the trapping potentials, we present a precise analysis on the concentration behavior of the minimizers as the coupling coefficient goes to a critical value, where the minimizers blow up and the maximum point sequence concentrates at a global minima of the associated trapping potentials. We also identify an optimal blowing up rate under polynomial potentials by establishing some delicate estimates of energy functionals.