I've had similar problem with add-on helicopters. I'm not a chopper pilot in real life and my knowledge may be sketchy, but as far as I know, the helicopter throttle needs to be set at or near max RPM and left there. The vertical and forward speed is then controlled with the collective only, ie the pitch of the blades. Some helicopters have a governor, which automatically adjusts throttle to pitch and speed but it will still need a full throttle setting to start off with. But you probably know all of that and are just struggling with the engines cutting out, right?.
Now, here is what I found with some add-on helicopters. Even if you increase the throttle as much as you can with your mapped joystick control or F4 key, the engines still cut out when you try to lift off. I found by pure coincidence that the throttle lever(s) in the cockpit needs another nudge with the mouse to lock into full power. If you're flying with 3D or virtual cockpit, this step will make your life a lot easier. Look around until you find the throttle lever(s) and click (and hold) on them with your mouse and physically move/drag or rotate them with mouse scroll wheel until they move fully forward. Most of the time you will see the motion. Once you're at full RPM, leave the throttle alone! and only fly with collective from there on as the engine needs consistent max RPM to deliver enough power to the blades once you start changing the pitch for lift off etc.
Hope that works for you, it solved my problems 9 out of 10 times.
i7-4790K @ 4.6 GHz, ASUS Maximus VII Gene, GTX 1070 8GB, 16GB DDR3 2400MHz, ASUS PG348Q 34" 3440x1440 monitor, CH Products Fighterstick, Pro Throttle and Pro Pedals, Track IR 5, Oculus Rift, Windows 7 64-bit