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.