Even if you do not believe it, you do not know. It is not clear if Jupiter, the largest planet in our solar system that has been lacking much to be a second star and whose magnetic field protects us from many of the “inclemencies” outside our system (whether radiation, whether Solid objects, by deflecting them by their gravity or their magnetic field) has a solid nucleus or if it is a plasma nucleus (like our star).
While we know that its atmosphere is mostly composed (helium and hydrogen, like any star), Jupiter still incorporates a lot of mystery in itself and The Juno spacecraft (NASA) is going to try to give a little more light regarding its atmosphere and, you will see if it is possible that you have nucleus or not measuring with more accuracy than your Voyager probes magnetosphere.
What is clear from the observations is that it has very important winds on its surface, which is dense as a soup, which must have convective currents due to its own gravity and the radiation that comes from the Sun (with what inside Of helium and liquid hydrogen, at least), that in its center must make “very hot” since it has not released all that it has (not like other planets that have no seismic activity and are therefore “cold” inside), which possesses an enormous amount of satellites and asteroids that in itself can be considered as another solar system (in miniature), that has rings (yes, like Saturn).
Summarizing: no, we do not know if it has a solid core or if it has a core as it is usually defined (central sphere composed of heavy materials, fundamentally magnetic that can be in the form of solid or liquid and that rotates) and, the only way To have an idea is to check the density of its atmosphere to be able to calculate through its magnetosphere if the only form that is generated and has such “size” is that there is this area.