A constrained minimization problem related to two coupled pseudo-relativistic Hartree equations

TL;DR

This paper investigates a constrained minimization problem for a system of two coupled pseudo-relativistic Hartree equations, establishing the existence of minimizers under certain conditions, thus extending previous results in the field.

Contribution

The paper generalizes prior work by proving the existence of minimizers for a broader class of parameters and potentials in coupled pseudo-relativistic Hartree equations.

Findings

Existence of minimizers for the energy functional under certain conditions.

Generalization of previous results to more general parameters and potentials.

Extension of the theoretical framework for coupled pseudo-relativistic Hartree systems.

Abstract

We are concerned with the following constrained minimization problem: $$e(a_{1},a_{2},\beta) := \inf\left\{E_{a_{1},a_{2},\beta}(u_{1},u_{2}): \|u_{1}\|_{L^{2}(\mathbb{R}^{3})} = \|u_{2}\|_{L^{2}(\mathbb{R}^{3})} = 1\right\},$$ where $E_{a_{1},a_{2},\beta}$ is the energy functional associated to two coupled pseudo-relativistic Hartree equations involving three parameters $a_{1}, a_{2}, \beta$ and two trapping potentials $V_1(x)$ and $V_2(x)$. In this paper, we obtain the existence of minimizers of $e(a_{1},a_{2},\beta)$ for possible $a_{1}, a_{2}$ and $\beta$ under suitable conditions on the potentials, which generalizes the results of the papers [16,17,18] in different senses.