We consider spherically symmetric motions of a polytropic gas under the self-gravitation governed by the Euler--Poisson equations. The adiabatic exponent ($=$ the ratio of the specific heats) $\gamma$ is assumed to satisfy $6/5 < \gamma \leq 2$. Then there are equilibria touching the vacuum with finite radii, and the linearized equation around one of the equilibria has time-periodic solutions. To justify the linearization, we should construct true solutions for which this time-periodic solution plus the equilibrium is the first approximation. We solve this problem by the Nash--Moser theorem. The result will realize the so-called physical vacuum boundary. But the present study restricts $\gamma$ to the case in which $\gamma/(\gamma-1)$ is an integer. Other cases are reserved to the future as an open problem. The time-local existence of smooth solutions to the Cauchy problems is also discussed.
"On spherically symmetric motions of a gaseous star governed by the Euler--Poisson equations." Osaka J. Math. 52 (2) 545 - 581, April 2015.