Good evening friend, you already ruled out that the source is ok because the monitor turns on, that means that at least two stages are working well that they are the source of power and the logic card, so for that fault you have several possible causes. 1. As explained by friend Cristian, it may be that the line that brings the ignition pulse signal from the video card is interrupted and therefore does not turn on the inverter. 2. It may be that the fuse (if any) or a resistance of the 12-14 volt line that goes to the inverter is open and the voltage does not reach, usually burns because there is a short in the inverter, a capacitor damaged in said circuit or simply left due to an overload of energy. 3. The short of which I speak can be an associated diode or the inverter msofet. 4. The inverter controller IC is damaged or has defective welds on its pins or some of its nearby components such as resistors or surface mount capacitors. 5. And this would be the worst case, both lamps are burned or broken, when only one is damaged, the circuit is completed and turns on giving luminance for a brief moment, but as the system detects the fault in one of the Lamps automatically protect and turn off leaving the screen completely dark. Take it easy and go checking step by step, luck, greetings.