battery and capicitor problems

The cap never turns off, but it leaks little current when the amps are off. Im assuming you have a stock alternator...? Your alternator cannot keep up with the current you are drawing and the cap is not helping. When your amps draw a good deal of current it is pulled from the line (some from the battery some from the cap) this then causes the alternator to charge the battery while the cap and amps are drawing current as well. Bottom line.... You need an alternator that can provide more current. Im assuming that your alternator is in good shape currently, it is possible that your alternator is not working properly, either way you need to replace it now or when it becomes damaged from the overdraw. Or turn it down...

Before you replace your alternator have your battery checked.  You could have a dead cell in your battery which causes it to die when place under a load.  I am not sure what you mean that your cap remains on, does you cap have a digital voltage meter on it that lights up and stays on.  If so this is not normal, they normally cut off after a few minutes of low to no current activity. 

If the average current draw of the audio system exceeds the maximum output of the alternator by a large amount, then it might be time to consider an alternator upgrade. A simple test is to start the vehicle, turn on the most commonly used vehicle accessories, then crank the audio system to full volume just under clipping. If the average battery voltage is above 13 volts, then the battery will receive sufficient charge from the alternator. No need to replace the alternator here. Assuming that the average current draw of the system is too high for the OEM charging system and the average battery voltage drops below 13 volts, it would make sense to install a high output alternator rather than extra batteries. Remember that when the vehicle is running, the battery is a load to the alternator.

Keep in mind that the high output alternator should have more output than the OEM alternator and high engine speeds, but may have less output than the OEM alternator at idle speed. If the vehicle will be driven in heavy stop and go traffic, the battery may not receive a sufficient charge.