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 Iα,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║2H1/2(R3) by the homogeneous one ║u║Ḣ1/2(R3) in the energy above.
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