  We search for self-gravitating oscillating field lumps (pulsons) in the
scalar model with logarithmic potential. With the use of a
Krylov-Bogoliubov-type asymptotic expansion in the gravitational constant, the
pulson solutions of the Einstein-Klein-Gordon system are obtained in the
Schwarzschild coordinates. They are expressed in terms of solutions of the
singular Hill's equation. The masses of the obtained pulsons are calculated.
The initial conditions are found under which the pulson solutions become
periodic. These conditions are then used in direct numerical integration of the
Einstein-Klein-Gordon system. It is shown that they do evolve into a very
long-lived periodic pulson. Stability of the self-gravitating pulsons and their
possible astrophysical applications are briefly discussed.
