When the adapter light goes off when plugged into the laptop it is usually a sign that a component on the board has an electrical short (usually the GPU). The adapter shuts power to protect itself.  Unfortunately the majority of parts on your machine are soldered to the board so it will mean you need to replace the motherboard.  You could try removing some of the parts that you can remove, such as the memory and storage drives but my gut says that it will be the GPU.  You can try parts-people for a replacement board or contact Dell for an Out-of-warranty repair. 
My Alienware R4 laptop was closed in hibernation mode and the fans were running and it was pretty hot. I turned it off and now it won't turn back on. The AC adapters blue LED light turns off when I plug it into my computer. Tried a different AC adaptor from another laptop, it had a blue light which began flashing when I plugged it into my computer. I opened up the back of my laptop and disconnected the battery, held the power button down and reconnected the battery but that did not work. Tried plugging in the AC adaptor with the battery disconnected but got the same results, with the blue LED turning off as soon as I plugged it in. What does it mean if the LED indicator light on the AC adaptor turns off when you plug it into the laptop? What happened to my laptop? It worked fine all day and is now completely useless. I own a business and use this laptop for work also, so I need a fix ASAP. 
17 R4, will not turn on