Spherically Symmetric Solutions in F(R) Gravity and Gravitational
  Repulsion
  Spherically symmetric solutions in F(R) theories in astronomical systems with
rising energy density are studied. The range of parameters is established for
which the flat space-time approximation for the background metric is valid. For
the solutions in which the curvature scalar oscillates with large amplitude and
high frequency, found in our previous papers, it is shown that the analysis of
the Jeans instability is strongly modified. It is discovered that for large
astronomical objects modified gravity can be repulsive, so such objects shrink
forming relatively thin shells instead of quasi uniform bodies. This may
explain the formation of cosmic voids.
