Funnily enough, we don’t actually know if gas giants have solid cores or not! However, more planet formation theories do use some sort of rocky core (protoplanet). Once that core reaches a certain mass, it starts to accrete enough gas that the gas mass helps it accrete more gas–a process called runaway accretion. This can pump up a small core from ~10 times the mass of Earth to Jupiter size or larger!
There are also theories of direct fragmentation, where gas can collapse in on itself and form a planet directly without a core required. This pathway is more akin to stellar formation and requires very massive protoplanetary disks.
They are thought to have a small solid core, just that the vast majority of the planet is gas, just like the Sun when you get that much gas into a small area/volume the gravity crushes the atoms and molecules together making the core of a gas giant denser and may even transform hydrogen from a gas to a metal. https://youtu.be/b-gCfHXNIVc
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