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Existence of steady states in elastic media at small strains with diffusionof a solvent or fluid due to Fick's or Darcy's laws is proved by combiningusage of variational methods inspired from static situations with Schauder'sfixed-point arguments. In the plain variant, the problem consists in the forceequilibrium coupled with the continuity equation, and the underlying operatoris non-potential and non-pseudomonotone so that conventional methods are notapplicable. In advanced variants, electrically-charged multi-component flowsthrough an electrically charged elastic solid are treated, employing criticalpoints of the saddle-point type. Eventually, anisothermal variants involvingheat-transfer equation are treated, too.

Variational methods for steady-state Darcy/Fick flow in swollen and poroelastic solids