Ground states for semi-relativiscti SPS energy

We prove the existence of ground states for the semi-relativistic Schrodinger-Poisson-Slater energy I-alpha,I- beta(rho) = inf (u is an element of H 1/2 (R3) integral R3 vertical bar u vertical bar 2dx=rho) 1/2 parallel to u parallel to(2)(H1/2(R3)) + alpha integral integral(R3xR3) vertical bar u(x)vertical bar(2)vertical bar u(y)vertical bar(2)/vertical bar x - y vertical bar dxdy - beta integral(R3) vertical bar u vertical bar(8/3)dx alpha, beta > 0 and rho > 0 is small enough. The minimization problem is L-2 critical and in order to characterize the values alpha, beta > 0 such that I-alpha,I- beta(rho) > -infinity for every rho > 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 1/S parallel to phi parallel to(L8/3(R3))/parallel to phi parallel to(1/2)(H1/2(R3)) <= (integral integral(R3 x R3) vertical bar phi(x)vertical bar(2)vertical bar phi(y)vertical bar(2)/vertical bar x - y vertical bar dxdy)(1/8) for all phi is an element of C-0(infinity)(R-3). Besides, we show that similar compactness property fails if we replace the inhomogeneous Sobolev norm parallel to u parallel to(2)(H1/2(R3)) by the homogeneous one parallel to u parallel to(H1/2(R3)) in the energy above.