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 $$I^{\alpha,\beta}(\rho)=\inf_{\substack{u\in H^\frac 12(\R^3) \int_{\R^3}|u|^2 dx=\rho}} \frac{1}{2}\|u\|^2_{H^\frac 12(\R^3)} +\alpha\int\int_{\R^{3}\times\R^{3}} \frac{| u(x)|^{2}|u(y)|^2}{|x-y|}dxdy-\beta\int_{\R^{3}}|u|^{\frac{8}{3}}dx$$ $\alpha,\beta>0$ 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 $I^{\alpha, \beta}(\rho)>-\infty$ 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 $$\frac{1}{S}\frac{\|\varphi\|_{L^\frac 83(\R^3)}}{\|\varphi\|_{\dot H^\frac 12(\R^3)}^\frac 12}\leq \left (\int\int_{\R^3\times \R^3} \frac{|\varphi(x)|^2|\varphi(y)|^2}{|x-y|}dxdy\right)^\frac 18 $$ for all $\varphi\in C^\infty_0(\R^3)$. Eventually we show that similar compactness property fails provided that in the energy above we replace the inhomogeneous Sobolev norm $\|u\|^2_{H^\frac 12(\R^3)}$ by the homogeneous one $\|u\|_{\dot H^\frac 12(\R^3)}$.