  We consider the compressible Euler-Poisson equations for polytropes
$P(\rho)=K\rho^{\gamma}$ with $\gamma\in \left(\frac{6}{5},\frac{4}{3} \right]$
and the white dwarf stars. For $\gamma=\frac{4}{3},$ we show that there exists
a global weak solution for the spherically symmetric initial data with mass
less than the mass of the Lane-Emden stars (i.e. non-rotating polytropes). For
$\gamma\in \left(\frac{6}{5},\frac{4}{3} \right)$, we show the existence of
global weak solution for spherical symmetric initial data in an invariant set
containing a neighborhood of Lane-Emden stars. Moreover, the support of these
solution expands to infinity. As a corollary, this proves the strong
instability of the Lane-Emden stars for $\gamma\in \left(
\frac{6}{5},\frac{4}{3}\right] $. For $\gamma\in \left(\frac{6}{5},\frac{4}{3}
\right),$ our results provide the first example of expanding solutions. For
white dwarf stars, we prove that the solution cannot collapse if the mass of
initial data is less than the Chandrasekhar limit mass, which is the supremum
of the mass of the non-rotating white dwarf stars. Our proof strongly uses the
variational characterization of the Lane-Emden stars. First, we relate the best
constant of a Hardy-Littlewood type inequality with the mass of the Lane-Emden
stars with $\gamma=\frac{4}{3}$, which is further shown to equal the
Chandrasekhar limit mass. For $\gamma\in\left( \frac{6}{5},\frac{4}{3}\right)
$, we show that the Lane-Emden stars are minimizers of an energy-mass
functional subject to a Pohozaev type constraint. This is crucial in the
construction of the invariant set of expanding solutions.
