Existence and mass concentration of pseudo-relativistic Hartree equation

Abstract

In this paper, we investigate the constrained minimization problem \begin{equation}\label{eq:0.1} e(a):=\inf_{\{u\in \mathcal{H},\|u\|_2^2=1\}}E_a(u), \end{equation} where the energy functional \begin{equation} \label{eq:0.2} E_a(u)=\int_{\mathbb{R}^3}(u\sqrt{-Δ+m^2}\,u+Vu^2)\,dx -\frac{a}{2}\int_{\mathbb{R}^3}(|x|^{-1}*u^2)u^2\,dx \end{equation} with $m\in \mathbb{R}$, $a>0$, is defined on a Sobolev space $\mathcal{H}$. We show that there exists a threshold $a^{\ast }>0$ so that $e(a)$ is achieved if $0<a<a^{\ast }$, and has no minimizers if $a\ge a^{\ast }$. We also investigate the asymptotic behavior of nonnegative minimizers of $e(a)$ as $a\to a^{\ast }$.

References in corpus (2)

• Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation
• On Blow-up Profile of Ground States of Boson Stars with External Potential

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• Ground states of a coupled pseudo-relativistic Hartree system: existence and concentration behavior
• Blow-up Profile of Neutron Stars in the Chandrasekhar theory
• Existence and non-existence of normalized solutions for a nonlinear fractional Schrödinger system
• Blow-up profile of neutron stars in the Hartree-Fock-Bogoliubov theory
• Boson Stars with Long-range Perturbations
• On supercritical nonlinear Schrödinger equations with ellipse-shaped potentials
• Normalized solutions and mass concentration for supercritical nonlinear Schrödinger equations