On spherically symmetric motions of a gaseous star governed by the Euler-Poisson equations

We study spherically symmetric motions of a gaseous star governed by the Euler-Poisson equations. Equilibria are given as solutions of the Lane-Emden equations, and the linearized equation around one of these equilibria admits time-periodic solutions. To justify the linearization, we should construct true solutions of the nonlinear evolution equation for which this time-periodic solutions plus the equilibrium is the first approximation. To do so, we apply the Nash-Moser theory. We presents a partial result under some assumptions on the adiabatic exponent of the gas, and propose an open problem to be studied in the future.