Ground states for the Hartree energy functional in the critical case

We consider the problem of finding a minimizer $u$ in $H^1({\mathbb{R}}^3)$ for the Hartree energy functional with convolution potential $w$ in $L^{\mathrm{\infty }}({\mathbb{R}}^3)+L^{3\mathrm{/}2,\mathrm{\infty }}({\mathbb{R}}^3)$ with $L^{\mathrm{\infty }}$ part vanishing at infinity. This class includes sums of potentials of the kind $-\frac{1}{\mathrm{\mid }x{\mathrm{\mid }}^{\alpha }}$, $0<\alpha \le 2$, together with the case $w$ in $L^{3\mathrm{/}2}({\mathbb{R}}^3)$. We prove the existence of such groundstates for a wide range of $L^2$ masses. We also establish basic properties of the groundstates, i.e.~positivity and regularity. Lastly, we exploit the estimates we derived for the stationary problem to prove global well-posedness of the associated evolution problem and orbital stability of the set of ground states.