On Blow-up Profile of Ground States of Boson Stars with External Potential

Abstract

We study minimizers of the pseudo-relativistic Hartree functional $${\mathcal{E}}_a(u):=\Vert (-\Delta +m^2{)}^{1/4}u{\Vert }_{L^2}^2-\frac{a}{2}{\int }_{{\mathbb{R}}^3}({\left|\cdot \right|}^{-1}\star |u{|}^2)(x)|u(x){|}^2\mathrm{d}x+{\int }_{{\mathbb{R}}^3}V(x)|u(x){|}^2\mathrm{d}x$$ under the mass constraint ${\int }_{{\mathbb{R}}^3}|u(x){|}^2\mathrm{d}x=1$. Here $m>0$ is the mass of particles and $V\ge 0$ is an external potential. We prove that minimizers exist if and only if $a$ satisfies $0\le a<a^{\ast }$, and there is no minimizer if $a\ge a^{\ast }$, where $a^{\ast }$ is called the Chandrasekhar limit}. When $a$ approaches $a^{\ast }$ from below, the blow-up behavior of minimizers is derived under some general external potentials $V$. Here we consider three cases of $V$: trapping potential, i.e. $V\in L_{\mathrm{loc}}^{\infty }({\mathbb{R}}^3)$ satisfies ${\lim }_{|x|\to \infty }V(x)=\infty$; periodic potential, i.e. $V\in C({\mathbb{R}}^3)$ stisfies $V(x+z)=V(x)$ for all $z\in {\mathbb{Z}}^3$; and ring-shaped potential, e.g. $V(x)=||x|-1{|}^p$ for some $p>0$.

To appear in Journal of Statistical Physics

References in corpus (1)

• Existence and mass concentration of pseudo-relativistic Hartree equation

Cited by in corpus (8)

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• Many-Body Blow-Up Profile of Boson Stars with External Potentials
• Blow-up Profile of Neutron Stars in the Chandrasekhar theory
• Ground states of a coupled pseudo-relativistic Hartree system: existence and concentration behavior
• Blow-up profile of neutron stars in the Hartree-Fock-Bogoliubov theory
• Boson Stars with Long-range Perturbations