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

Abstract

We study minimizers of the pseudo-relativistic Hartree functional $$\mathcal{E}_{a}(u):=\|(-\Delta+m^{2})^{1/4}u\|_{L^{2}}^{2}-\frac{a}{2}\int_{\mathbb{R}^{3}}(\left|\cdot\right|^{-1}\star |u|^{2})(x)|u(x)|^{2}{\rm d}x+\int_{\mathbb{R}^{3}}V(x)|u(x)|^{2}{\rm d}x$$ under the mass constraint $\int_{\mathbb{R}^3}|u(x)|^2{\rm d}x=1$. Here $m>0$ is the mass of particles and $V\geq 0$ is an external potential. We prove that minimizers exist if and only if $a$ satisfies $0\leq a<a^{*}$, and there is no minimizer if $a\geq a^*$, where $a^*$ is called the Chandrasekhar limit}. When $a$ approaches $a^*$ 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. $V\in L_{{\rm loc}}^{\infty}(\mathbb{R}^3)$ 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 $p>0$.