Poincaré's paper is

H. Poincaré, *Sur une theoreme de M. Liapunoff relatif 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).

As discussed by <A HREF="http://www.ams.org/journals/bull/1941-47-10/S0002-9904-1941-07541-5/S0002-9904-1941-07541-5.pdf">G.C. Evans,</A>
Poincaré assumes tacitly that there do exist one or more bodies of a given volume, with smooth boundaries, which provide relative minima for the Coulomb energy with respect to neighboring forms; and his treatment amounts to a proof that among these the sphere provides the absolute minimum.