Well-Posedness for Semi-Relativistic Hartree Equations of Critical Type

We prove local and global well-posedness for semi-relativistic, nonlinear Schr"odinger equations $i{\mathrm{\partial }}_{t}u=\sqrt{-\mathrm{\Delta }+m^2}u+F(u)$ with initial data in $H^s({\mathbb{R}}^3)$, $s\ge 1\mathrm{/}2$. Here $F(u)$ is a critical Hartree nonlinearity that corresponds to Coulomb or Yukawa type self-interactions. For focusing $F(u)$, which arise in the quantum theory of boson stars, we derive a sufficient condition for global-in-time existence in terms of a solitary wave ground state. Our proof of well-posedness does not rely on Strichartz type estimates, and it enables us to add external potentials of a general class.