Incorporating boundary conditions at infinity into simulations on bounded computational domains is a repeatedly occurring problem in scientific computing. The combination of finite element methods (FEM) and boundary element methods (BEM) is the obvious instrument, and we adapt here for the first time the two standard FEM-BEM coupling approaches to the free-boundary equilibrium problem: the Johnson-Nédélec coupling and the Bielak-MacCamy coupling. We recall also the classical approach for fusion applications, dubbed according to its first appearance von-Hagenow-Lackner coupling and present the less used alternative introduced by Albanese, Blum and de Barbieri in [2]. We show that the von-Hagenow-Lackner coupling suffers from undesirable non-optimal convergence properties, that suggest that other coupling schemes, in particular Johnson-Nédélec or Albanese-Blum-de Barbieri are more appropriate for non-linear equilibrium problems. Moreover, we show that any of such coupling methods requires Newton-like iteration schemes for solving the corresponding non-linear discrete algebraic systems.