It looks like you have the "memory slot problem"

IMPORTANT! Take out the battery, and leave it out for the following tests.

Unplug the AC power. Open the RAM bay on the bottom of the machine (it's a hatch about 2" square with a single screw). The SODIMMs are held in by springs on either end that you press outward to release the module. After inserting a module, press down on it until the springs catch.

• If you have two memory modules, try each of them singly in both slots. If either module works reliably in both slots, the other module is probably defective. Buy a new module.
  
  Be sure to unplug the power after each test.

• If you have one memory module, try it in the other slot.
  
  
• If one slot (or both) is consistently dead with any module plugged into it, we can probably fix it. Go to shipping.
  
  
• A further diagnostic is to press the memory slot towards the main board (use something non-metallic and insulated, please) while the machine is powering up. If it sometimes boots at least as far as the Vaio splash screen, your odds of a successful repair increase to 99.9%.

IMPORTANT! If your computer exhibits any other light pattern than the following, we can't fix it:

• At power-on, the (green) power light comes on and stays on.
• The (yellow) HDD light flickers and goes out.
• There is no image on the screen, and no sound from the speaker.