This is an interesting fight. Lavos Core is made up of the central core and left and right bits. The main objective in this fight isn't destroying the central core, even though it would seem that way. The part with the most hp is the right part. First destroy the left bit and eventually the cores defense will drop. Now is the time to finish off the center. His hp isn't too drasticly high. While being well protected and heal throughout the fight, finaly it's time to finish this boss (Or is it??). Unless you are extremley powerful, the right core will revive it's partners before you defeat it. When they return they will have less hp so defeat them quickly and continue at you objective. They may be revived more than once, but it's not always nesascary to re-kill them when you can continue your onslaught on the right core. When you've depleted the right cores hp the remaining core part will also vanish, and congratulations you're done!
After completing the quest for the rainbow shell and youve obtained the sun stone, go to the basement in the year 1000 A.D. Where Melchior should be. If you haven't already choose between the armour choices he gives you, then talk to him again and he will make Cronos ultimate weapon "Rainbow" and Sun Specs.