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

TL;DR

This work analyzes ground states for a coupled pseudo-relativistic Hartree system in $\mathbb{R}^3$ with trapping potentials and attractive intra- and interspecies interactions. It establishes a complete existence/nonexistence classification for the constrained minimization problem, identifies a critical coupling $\beta^*$ that governs blow-up, and proves that minimizers concentrate at global minima of the potentials as $\beta\to\beta^*$. In the general potential setting, the authors show that rescaled minimizers converge to weighted copies of the one-component ground state $Q$, with the concentration point approaching a common zero of the potentials. For polynomial potentials, they obtain an explicit blow-up rate and energy asymptotics, revealing that concentration occurs at flattest common minima and providing precise scaling laws. The analysis combines nonlocal De Giorgi–Nash–Moser techniques with refined energy estimates to handle the double nonlocality from the pseudo-relativistic operator and the Hartree nonlinearity, yielding sharp concentration behavior results with practical implications for multi-component relativistic Bose–Einstein–type systems.

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.

Theorems & Definitions (19)

• Theorem 1.1
• Remark 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• Lemma 3.1