Uniqueness of ground states for the L^2-critical boson star equation

We establish uniqueness of ground states $u(x)\ge 0$ for the $L^2$-critical boson star equation $\sqrt{-\mathrm{\Delta }}u-(\mathrm{\mid }x{\mathrm{\mid }}^{-1}\ast \mathrm{\mid }u{\mathrm{\mid }}^2)u=-u$ in ${\mathbb{R}}^3$. The proof blends variational arguments with the harmonic extension to the halfspace ${\mathbb{R}}_{+}^4$. Apart from uniqueness, we also show radiality of ground states (up to translations) and the nondegeneracy of the linearization. Our results provide an indispensable basis for the blowup analysis of the time-dependent $L^2$-critical boson star equation. The uniqueness proof can be generalized to different fractional Laplacians $(-\mathrm{\Delta }{)}^s$ and space dimensions.