Abstract: The liquid Lane-Emden star is a free boundary problem of compressible Euler-Poisson equation which describes motion of celestial bodies. This model admits a class of static solutions parametrized by its central density. According to Lam [9], when the central density is sufficiently small or the adiabatic constant $\gamma\in [\frac43,2]$, the static solutions are linearly stable. In this article, by constructing periodic solutions to the linearized equation, we prove that even though these solutions are linearly stable, they may not decay in time. Moreover we prove that if the sum of the internal energy and potential energy of this model has an minimizer, then it must be the spherically symmetric solution to the static equation, therefore demonstrating their stability from a variational point of view.

Key words: compressible liquid, static solution, uniqueness