How does earth core remain molten?
primarily from the energy generated by nuclear decay of radioactive substances in the core of the earth.
Lord Kelvin calculated the maximum age of the earth from cooling at the surface of the earth. Lord Kelvin calculated this age from the assumption that the only source of heat was the initial formation of the earth to be about 20 million years This age is thought to be much to young. The young age of the earth based on ambient cooling was not sufficient for the popular theory of Darwinian evolution. Therefore it was postulated that there must another source of heat energy in the earth.
When radioactivity was discovered and the immense amount of energy that can be generated through radioactive decay, the age of the earth was extended backwards to match the half-life of Uranium 238. The amount of energy produced by radioactive decay is based on the famous equation
There is no way to measure the amount of radioactive decay in the core of the earth. The idea that the heat energy of the earth comes from radioactive decay is purely theoretical. It is based on the assumption of a vast age for the earth ( 4.5 billion years) and the fact that ambient cooling of the earth's initial energy does not account for such vast ages.