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The damaging effect of strong electron-electron repulsion on regular,electron-phonon %$s$-wave superconductivity is a standard tenet. In spite ofthat, an increasing number of compounds such as fullerides and more recentlyalkali-doped aromatics exhibit %$s$-wave or presumably $s$ wavesuperconductivity despite very narrow bands and very strong electron repulsion.Here, we explore superconducting solutions of a model Hamiltonian inspired bythe electronic structure of alkali doped aromatics. The model is a two-site,two-narrow-band metal with a single intersite phonon, leading toattraction-mediated, two-order parameter superconductivity. On top of that, themodel includes a repulsive on-site Hubbard $U$, whose effect on thesuperconductivity we study. Starting within mean field, we find that $s \pm$superconductivity is the best solution surviving the presence of $U$, whoseeffect is canceled out by the opposite signs of the two order parameters. Thecorrelated Gutzwiller study that follows is necessary because without electroncorrelations the superconducting state would in this model be superseded by anantiferromagnetic insulating state with lower energy. The Gutzwillercorrelations lower the energy of the metallic state, with the consequence thatthe $s \pm$ superconducting state is stabilized and even strengthened for smallHubbard $U$.

Robust $s\pm$ Superconductivity in a Two-Band Hubbard-Fr{\"o}hlich Model of Alkali Doped Organics