May sound silly, but does the CPU fan whir when you power on the system? I had a similar problem with a Dell Inspiron 5160 and all it took was a CPU fan replacement. Most portable devices, chiefly laptops, have an automatic shut off feature if it "feels" the CPU is too hot and usual event horizon is right after the POST (just before or during the operating system load screen). It's more tedious than working on a desktop but Dell laptops are fairly easy to work inside assuming you know how to replace the heat-sink/fan assembly. You can find new and used kits which work almost identically in most cases and include the heat-sink and fan for relatively cheap online. Just be sure you get the exact same heat-sink model and voltage fan assembly for your laptop.