On ground states for the L²-critical boson star equation

We consider ground state solutions $u \geq 0$ for the $L^2$-critical boson star equation $$ \sqrt{-\Delta} \, u - \big (|x|^{-1} \ast |u|^2 \big) u = -u \quad {in $\R^3$}. $$ We prove analyticity and radial symmetry of $u$. In a previous version of this paper, we also stated uniqueness and nondegeneracy of ground states for the $L^2$-critical boson star equation in $\R^3$, but the arguments given there contained a gap. However, we refer to our recent preprint \cite{FraLe} in {\tt arXiv:1009.4042}, where we prove a general uniqueness and nondegeneracy result for ground states of nonlinear equations with fractional Laplacians in $d=1$ space dimension.