On nonlinear instability of liquid Lane-Emden stars

We establish a dynamical nonlinear instability of liquid Lane-Emden stars in ${\mathbb{R}}^3$ whose adiabatic exponents take values in $[1,\frac{4}{3})$. Our proof relies on a priori estimates for the free boundary problem of a compressible self-gravitating liquid, as well as a quantitative analysis of the competition between the fastest linear growing mode and the source.