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

We study minimizers of the pseudo-relativistic Hartree functional $${\mathcal{E}}_a(u):=\mathrm{\parallel }(-\mathrm{\Delta }+m^2{)}^{1\mathrm{/}4}u{\mathrm{\parallel }}_{L^2}^2-\frac{a}{2}{\int }_{{\mathbb{R}}^3}({\mid \cdot \mid }^{-1}\star \mathrm{\mid }u{\mathrm{\mid }}^2)(x)\mathrm{\mid }u(x){\mathrm{\mid }}^2\mathrm{d}x+{\int }_{{\mathbb{R}}^3}V(x)\mathrm{\mid }u(x){\mathrm{\mid }}^2\mathrm{d}x$$ under the mass constraint ${\int }_{{\mathbb{R}}^3}\mathrm{\mid }u(x){\mathrm{\mid }}^2\mathrm{d}x=1$. 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. $V\in L_{\mathrm{l}\mathrm{o}\mathrm{c}}^{\mathrm{\infty }}({\mathbb{R}}^3)$ satisfies ${\lim }_{\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }}V(x)=\mathrm{\infty }$; periodic potential, i.e. $V\in C($\mathbb{R}$^3)$stisfies$V(x+z)=V(x)$forall$z\in$\mathbb{Z}$^3$; and ring-shaped potential, e.g. $V(x)=\mathrm{\mid }\mathrm{\mid }x\mathrm{\mid }-1{\mathrm{\mid }}^p$ for some $p>0$.