  We obtain two equations (following from two different approaches) for the
density profile in a self-gravitating polytropic cylindrically symmetric and
rotating turbulent gas disk. The adopted physical picture is appropriate to
describe the conditions near to the cloud core where the equation of state of
the gas changes from isothermal (in the outer cloud layers) to one of "hard
polytrope", and the symmetry changes from spherical to cylindrical. On the
assumption of steady state, as the accreting matter passes through all spatial
scales, we show that the total energy per unit mass is an invariant with
respect to the fluid flow. The obtained equation describes the balance of the
kinetic, thermal and gravitational energy of a fluid element. We also introduce
a method for approximating density profile solutions (in a power-law form),
leading to the emergence of three different regimes. We apply, as well,
dynamical analysis of the motion of a fluid element. Only one of the regimes is
in accordance with the two approaches (energy and force balance). It
corresponds to a density profile of a slope -2, polytropic exponent 3/2, and
sub-Keplerian rotation of the disk, when the gravity is balanced by the thermal
pressure. It also matches with some observations and numerical works and, in
particular, leads to a second power-law tail (of a slope approx. -1) of the
density distribution function in dense, self-gravitating cloud regions.
