  The Choquard equation is a partial differential equation that has gained
significant interest and attention in recent decades. It is a nonlinear
equation that combines elements of both the Laplace and Schr\"odinger
operators, and it arises frequently in the study of numerous physical
phenomena, from condensed matter physics to nonlinear optics.
  In particular, the steady states of the Choquard equation were thoroughly
investigated using a variational functional acting on the wave functions.
  In this article, we introduce a dual formulation for the variational
functional in terms of the potential indiced by the wave function, and use it
to explore the existence of steady states of a multi-state version the Choquard
equation in critical and sub-critical cases.
