On uniqueness of the static state for a general compressible fluid

The system describing a thermodynamic equilibrium of a general compressible fluid in a potential field is studied. It is shown that there is a unique solution of this system in the class {ϑ>0;ϱ∈L1(Ω),ϱ⩾0a.e.} A general physically relevant constitutive function for the pressure is considered assuming just two additional conditions, namely that the pressure grows in ϱ at least as ϱ1-1/N and it behaves like a linear function as ϱ→0.