The relevant references are given on page 3 of <A HREF="https://arxiv.org/abs/1512.07282">arXiv:1512.07282</A>. 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).