= Solution
Let $M(r)$ be the <enclosed mass>. A spherical shell of thickness $dr$ has inward gravitational acceleration $GM(r)/r^2$, so <hydrostatic pressure support equation>[hydrostatic equilibrium] requires
$$
\boxed{\frac{dP}{dr}=-\frac{GM(r)\rho(r)}{r^2},
\qquad
\frac{dM}{dr}=4\pi r^2\rho(r).}
$$
With $\rho=Ar^2P$, these equations reduce to
$$
P'=-GAMP,
\qquad
M'=4\pi Ar^4P.
$$
For a regular central value $P(0)=P_c>0$, the first equation integrates to
$$
P(r)=P_c\exp\left(-GA\int_0^rM(s)\,ds\right).
$$
The <exponential function> is strictly positive at every finite $r$, so the pressure and density cannot reach zero at a finite stellar surface. The equations may describe an atmosphere whose density decreases at large radius, but they admit no nontrivial hydrostatic star of finite radius. Therefore
$$
\boxed{\text{there is no stable finite-radius stellar solution with this relation.}}
$$
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