On nonlinear instability of liquid Lane–Emden stars

We establish a dynamical nonlinear instability of liquid Lane–Emden stars in \({\mathbb {R}}^{3}\) ${\mathbb{R}}^3$ whose adiabatic exponents take values in \([1,\frac{4}{3})\) $[1,\frac{4}{3})$ . 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.