Existence and nonlinear stability of stationary states for the semi-relativistic Schr\"odinger-Poisson system

We study the stationary states of the semi-relativistic Schr\"odinger-Poisson system in the repulsive (plasma physics) Coulomb case. In particular, we establish the existence and the nonlinear stability of a wide class of stationary states by means of the energy-Casimir method. We generalize the global well-posedness result of our previous work for the semi-relativistic Schr\"odinger-Poisson system to spaces with higher regularity.