Well-posedness for semi-relativistic Hartree equations of critical type

We prove local and global well-posedness for semi-relativistic, nonlinear Schrödinger equations i∂tu=−Δ+m2 √u+F(u) with initial data in Hs(ℝ3), s⩾1/2. 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 global-in-time existence for small initial data, where the smallness condition is expressed in terms of the L2-norm of solitary wave ground states. Our proof of well-posedness does not rely on Strichartz type estimates. As a major benefit from this, our method enables us to consider external potentials of a quite general class.