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

Abstract

We study minimizers of the pseudo-relativistic Hartree functional ${\mathcal {E}}_{a}(u):=\Vert (-\varDelta +m^{2})^{1/4}u\Vert _{L^{2}}^{2}+\int _{{\mathbb {R}}^{3}}V(x)|u(x)|^{2}\mathrm{d}x-\frac{a}{2}\int _{{\mathbb {R}}^{3}}(\left| \cdot \right| ^{-1}\star |u|^{2})(x)|u(x)|^{2}\mathrm{d}x$ 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 $0\le a<a^{*}$, and there is no minimizer if $a\ge a^*$, 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. $V\in L_{\mathrm{loc}}^{\infty }({\mathbb {R}}^3)$ satisfies $\lim _{|x|\rightarrow \infty }V(x)=\infty$; periodic potential, i.e. $V\in C({\mathbb {R}}^3)$ satisfies $V(x+z)=V(x)$ for all $z\in \mathbb {Z}^3$; and ring-shaped potential, e.g. $V(x)=||x|-1|^p$ for some $p>0$.