Hi,
Even i had the same issue. This is a common problem not only for gateway but for other models also. Mostly this problem occurs due to overheating caused by using the laptops without proper bottom ventilation, like when up keep it above bed, pillow or so. Overheating usually blows up the graphics card, by which normally we have to replace the motherboard. I hope the laptops you've mentioned is not brand new. They never happen in them.
If the laptop is under warranty get the motherboard replaced. Else i would recommend you to approach a technician for the part replacement, if you are not well aware of the hardware part.
Gateway Manual to disassemble if you are an expert.
Thanx :)

A. Shut down the laptop and disconnect it from power cord. B. Remove battery after turn it over. C. Hold down the power button for 30 seconds. D Connect the power cord and try to turn it on. E. Now, at the start up try F8 and choose safe mode with networking. F. Else, at the startup press F2(Delete or F1) to go to BIOS and set BIOS to default settings. Hit F10, ESC once and choose Save and Exit and hit enter. G. If all of this does not work, you can try to reset your computer to factory setting by inserting the gateway recovery disks. Note: This process would delete all of the data of the computer. H.If this all also does not work, please try to reinstall the operating system. I.If this does not work, you can visit http://support.gateway.com.
Enter your product information and download the user manual.

Sometimes this problem occurs on gateways, due to the power converter for the LCD display. Since there is connected the LCD the signal isn't going at the external monitor. I suggest you open the case and unplug the internal vga cable in order to check the external monitor, and try to find power converter for the LCD as a replacement (maybe you should try change over between the 3 pieces) In case of a problem or clarification, don't hesitate to post me a reply. Thanks and regards Please kindly rate this solution Stelios direct FixYa link: http://www.fixya.com/users/technical114

If either lcd ner external vga do not light up...and you hear the windows logging sound...it means that the video card is out ..and you have to open it ..and see whats going on ..it is either a loose plug or wire or is dead for good..and need replacement ...
another issue if it sounds like a loop booting ..is the ram memory (in this case it do not boot to windows at all and you do not hear the sound..) and it makes a click like sound each time it boots ( a few seconds apart )
In this case remove the power and battery ...get the ram memory out and place it back ...start it again and see what happens..
also try to open it up ..remove the keyboard ..and remove the bios battery for a few seconds..it may have a odd setting that disabled the video card or so ...and this will default it ..
if non of this work ...it is the video card for sure ..but ...try first to replace the ram memory from one that works ...cause if the ram memory is out ...it behaves like this too..cause the ram memory is the first thing a computer sees..
meanwhile I will look for a scheme or something ..but I dont think you need one ..cause things are simple ...better a video to see how you have to open it ..inside they are
pretty compact and on modules...not too many wires so ..you can make anything wrong..
and for Ram memory you do not need to open it ..but just remove the back lid..
http://support.gateway.com/s/Manuals/Mobile/8512055_Oasis_RPL.pdf