since death star is capable of delivery a blast with high energy density, its core might be a nuclear fusion or anti matter power plant. maybe the mass there generates sufficient gravitational force.
The main weapon of death star was powered by kyber crystals, the same crystals that powers a Jedi's lighsaber.
That said, the death star also needs power for various other stuff not only the planet destroying mega cannon, so it's easy to assume there's also a powerful power plant for powering rest of the station.
There was also 123 "highly radioactive" generators powering it's propulsion system and hyperdrive.