Try removing the battery and then boot up using the ac adapter. See if the keyboard lights flash and if the Dell splash screen appears. If the system turns off immediately, the problem is a bad motherboard. It may be getting a false indication of over heating which will turn the system off to protect the processor. It also could be a power management problem. The over heat detection and power management are built into the motherboard.
If you get a beeping noise, count the beeps and let us know. There are several beep codes that indicate what has failed. This would indicate that the motherboard is good but a component such as memory has failed.