Ground states for semi-relativistic Schr\"odinger-Poisson-Slater energies

We prove the existence of ground states for the semi-relativistic Schr"odinger-Poisson-Slater energy $$I^{\alpha ,\beta }(\rho )=\underset{\begin{array}{c}u\in H^{\frac{1}{2}}({\mathbb{R}}^3){\int }_{{\mathbb{R}}^3}\mathrm{\mid }u{\mathrm{\mid }}^{2}dx=\rho \end{array}}{\inf }\frac{1}{2}\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{\frac{1}{2}}({\mathbb{R}}^3)}^2+\alpha \int {\int }_{{\mathbb{R}}^3\times {\mathbb{R}}^3}\frac{\mathrm{\mid }u(x){\mathrm{\mid }}^2\mathrm{\mid }u(y){\mathrm{\mid }}^2}{\mathrm{\mid }x-y\mathrm{\mid }}dxdy-\beta {\int }_{{\mathbb{R}}^3}\mathrm{\mid }u{\mathrm{\mid }}^{\frac{8}{3}}dx$$ $\alpha ,\beta >0$ and $\rho >0$ is small enough. The minimization problem is $L^2$ critical and in order to characterize of the values $\alpha ,\beta >0$ such that $I^{\alpha ,\beta }(\rho )>-\mathrm{\infty }$ 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 $$\frac{1}{S}\frac{\mathrm{\parallel }\phi {\mathrm{\parallel }}_{L^{\frac{8}{3}}({\mathbb{R}}^3)}}{\mathrm{\parallel }\phi {\mathrm{\parallel }}_{{\dot{H}}^{\frac{1}{2}}({\mathbb{R}}^3)}^{\frac{1}{2}}}\le {(\int {\int }_{{\mathbb{R}}^3\times {\mathbb{R}}^3}\frac{\mathrm{\mid }\phi (x){\mathrm{\mid }}^2\mathrm{\mid }\phi (y){\mathrm{\mid }}^2}{\mathrm{\mid }x-y\mathrm{\mid }}dxdy)}^{\frac{1}{8}}$$ for all $\phi \in C_0^{\mathrm{\infty }}({\mathbb{R}}^3)$. Eventually we show that similar compactness property fails provided that in the energy above we replace the inhomogeneous Sobolev norm $\mathrm{\parallel }u{\mathrm{\parallel }}_{H^{\frac{1}{2}}({\mathbb{R}}^3)}^2$ by the homogeneous one $|u|_{\dot H^\frac 12(\R^3)}$.