On the asymptotic decay of the Schr\"odinger--Newton ground state

The asymptotics of the ground state $u(r)$ of the Schr"odinger–Newton equation in ${\mathbb{R}}^3$ was determined by V. Moroz and J. van Schaftingen to be $u(r)\sim A{e}^{-r}\mathrm{/}{r}^{1-\mathrm{\parallel }u{\mathrm{\parallel }}_2^2\mathrm{/}8\pi }$ for some $A>0$, in units that fix the exponential rate to unity. They left open the value of $\mathrm{\parallel }u{\mathrm{\parallel }}_2^2$, the squared $L^2$ norm of $u$. Here it is rigorously shown that $2^{1/3}3\pi^2\leq |u|_2^2\leq 2^{3}\pi^{3/2}$\mathrm{.}Itisreportedthatnumerically$|u|_2^2\approx 14.03\pi$,revealingthatthemonomialprefactorof$e^{-r}$increaseswith$r$ in a concave manner. Asymptotic results are proposed for the Schr"odinger–Newton equation with external $\sim -K\mathrm{/}r$ potential, and for the related Hartree equation of a bosonic atom or ion.