  We present the results of our recent study on the interactions between a
giant planet and a self-gravitating gas disk. We investigate how the disk's
self-gravity affects the gap formation process and the migration of the giant
planet. Two series of 1-D and 2-D hydrodynamic simulations are performed. We
select several surface densities and focus on the gravitationally stable
region. To obtain more reliable gravity torques exerted on the planet, a
refined treatment of disk's gravity is adopted in the vicinity of the planet.
Our results indicate that the net effect of the disk's self-gravity on the gap
formation process depends on the surface density of the disk. We notice that
there are two critical values, \Sigma_I and \Sigma_II. When the surface density
of the disk is lower than the first one, \Sigma_0 < \Sigma_I, the effect of
self-gravity suppresses the formation of a gap. When \Sigma_0 > \Sigma_I, the
self-gravity of the gas tends to benefit the gap formation process and enlarge
the width/depth of the gap. According to our 1-D and 2-D simulations, we
estimate the first critical surface density \Sigma_I \approx 0.8MMSN. This
effect increases until the surface density reaches the second critical value
\Sigma_II. When \Sigma_0 > \Sigma_II, the gravitational turbulence in the disk
becomes dominant and the gap formation process is suppressed again. Our 2-D
simulations show that this critical surface density is around 3.5MMSN. We also
study the associated orbital evolution of a giant planet. Under the effect of
the disk's self-gravity, the migration rate of the giant planet increases when
the disk is dominated by gravitational turbulence. We show that the migration
timescale associates with the effective viscosity and can be up to 10^4 yr.
