Ground states for semi-relativistic Schrödinger-Poisson-Slater energy

We prove the existence of ground states for the semi-relativistic Schrödinger- Poisson-Slater energy (Formula presented) α, β > 0 and ρ > 0 is small enough. The minimization problem is L² critical and in order to characterize the values α, β > 0 such that I^α,β(ρ) > –∞ for every ρ > 0, we prove a new lower bound on the Coulomb energy involving the kinetic energy and the exchange energy. We prove the existence of a constant S > 0 such that (Formula presented) for all φ ∈ C₀^∞ (R³). Besides, we show that similar compactness property fails if we replace the inhomogeneous Sobolev norm ║u║²_H^1/2(R³) by the homogeneous one ║u║_Ḣ^1/2(R³) in the energy above.