The relevant references are given on page 3 of arXiv:1512.07282. Poincaré's paper is
H. Poincar ́e, Sur une th ́eorème de M. Liapunoff r ́elatif a l’ ́equilibre d’une masse fluide, Comptes Rendus de L’Academie des Sciences 104, 622–625 (1887).
The first complete proof is due to
E.H. Lieb, Existence and uniqueness of the minimizing solution of Choquard’s nonlinear equation, Studies in Appl. Math. 57, 93–105 (1977).