= Solution
Let $a_T^2=k_BT/(\mu m_u)$ be the constant <isothermal sound speed> squared. The stationary galactic gas obeys <hydrostatic equilibrium> in the specified <gravitational potential>:
$$
a_T^2\frac{d\rho}{dr}=-\rho\frac{d\psi}{dr},\qquad
\frac{d\ln\rho}{dr}=-\frac{1}{a_T^2}\frac{d\psi}{dr}.
$$
Integrating from the pressure-matching radius $r_0$ gives $\rho(r)=\rho(r_0)\exp[-(\psi(r)-\psi(r_0))/a_T^2]$. Since $\rho(r_0)=p_0/a_T^2$ and $\psi=-Ar^{-1/2}$,
$$
\boxed{\rho(r)=\frac{\mu m_up_0}{k_BT}
\exp\!\left[\frac{A\mu m_u}{k_BT}
\left(r^{-1/2}-r_0^{-1/2}\right)\right],\qquad0<r\leq r_0.}
$$
The exponent is dimensionless: the quantity in parentheses times $A$ is a difference of <gravitational potential>, of units $\mathrm{m^2\,s^{-2}}$. Density increases inward, as expected. The idealized potential is singular at the centre, so this formal profile should not be extrapolated into a region where the prescribed potential or the neglect of gas self-gravity fails.
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