Linear Stability of liquid Lane-Emden stars

We establish various qualitative properties of liquid Lane-Emden stars in ${\mathbb{R}}^d$, including bounds for its density profile $\rho$ and radius $R$. Using them we prove that against radial perturbations, the liquid Lane-Emden stars are linearly stable when $\gamma \ge 2(d-1)\mathrm{/}d$; linearly stable when $\gamma <2(d-1)\mathrm{/}d$ for stars with small relative central density $\rho (0)-\rho (R)$; and linearly unstable when $\gamma <2(d-1)\mathrm{/}d$ for stars with large central density. Such dependence on central density is not seen in the gaseous Lane-Emden stars.