Static black holes in equilibrium with matter: nonlinear equation of state

We consider a spherically symmetric black hole in equilibrium with surrounding classical matter that is characterized by a nonlinear dependence of the radial pressure p_{r} on the density {\rho}. We examine under which requirements such an equilibrium is possible. It is shown that if the radial and transverse pressures are equal (Pascal perfect fluid), equation of state should be approximately linear near the horizon. The corresponding restriction on ((dp_{r})/(d{\rho})) is a direct generalization of the result, previously found for an exactly linear equation of state. In the anisotropic case there is no restriction on equation of state but the horizon should be simple (nondegenerate).