Ground states for the Hartree energy functional in the critical case

Abstract

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 $L^\infty(\mathbb{R}^3)+L^{3/2,\infty}(\mathbb{R}^3)$ with $L^\infty$ part vanishing at infinity. This class includes sums of potentials of the kind $-\frac{1}{|x|^\alpha}$, $0<\alpha\le2$, together with the case $w$ in $L^{3/2}(\mathbb{R}^3)$. 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.