It's a mainboard failure -- and on a system that's now about 14 years old, it means it's time to start shopping for a replacement notebook.  
My Dell Studio 1535 is not turning on. I have the battery and the power cord plugged in but nothing is showing up on the screen as well as no lights or sounds. Nothing at all when plugged in. I tried removing the battery to see if that would work fix the problem but it didn't.
Dell Studio 1535 A/C power does not work