The most likely cause is that your battery cannot supply enough current at full throttle. It could be that the "C" rating of the battery is too low, the battery is not fully charged, or the wires going from the battery to the ESC are too small in diameter, too long, or too resistive.
It sounds like the voltage from the battery getting to the ESC is "Sagging" and the ESC low voltage cut out is being triggered!
Have fun!