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We obtain two equations (following from two different approaches) for the density profile in a self-gravitating polytropic cylindrically symmetric and rotating turbulent gas disc. 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 disc, 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 ∼−1) of the density distribution function in dense, self-gravitating cloud regions. [ABSTRACT FROM AUTHOR]