I quote from <A HREF="https://books.google.nl/books?id=YkUACwAAQBAJ&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false">Barry Simon, Harmonic Analysis: A Comprehensive Course in Analysis, Part 3</A> (page 197):

> That any positive harmonic function is constant is due to Bôcher
> (1902), although the theorem is often named after Picard’s rediscovery
> (1923) — there is often reference to the Liouville–Picard theorem.

Bôcher states the theorem in a footnote:

<IMG SRC="https://ilorentz.org/beenakker/MO/Liouville_1.png"/>

M. Bôcher, <A HREF="https://projecteuclid.org/euclid.bams/1183417472">*Singular points of functions which satisfy partial differential equations of the elliptic type*,</A> Bull. Amer. Math. Soc. (2) 9, (1903), 455–465. 

É. Picard, *Deux théorèmes élèmentaires sur les singularités des fonctions harmoniques*, C. R. Acad. Sci. Paris 176 (1923), 933–935.