[SpaceShouldBeMilitarized (OP)]

I think that the radiation and enormous gravity from the giant would make life impossible for it or it's moons. 

[Soul Surfer]

If there were a planet the size of jupiter orbiting where the earth is now and there was one large satellite the size and composition of the earth in a reasonably circular orbit around it and this orbit was around the distance of the galilean satellites of jupiter,  i.e. with a period around 1-10days and the rotation period was locked to the orbital period.  Complex Life could probably exist.  Life could also probably exist if the orbit around jupiter was much longer period and the planet had a rotation period  of a few days or less.  Departing from this would gradually increase the problems.  I am assuming that the "earth's" own magnetosphere would protect it from the worst of the radiation in "jupiter's"  magnetosphere but reasonably basic life can be quite radiation resistant  and you must remember that simple life even exists more than a mile below the surface in the earth's crust.