After uploading a script to my Leonardo and applying external power, the yellow L light starts immediately fading in/out and the script does not work. Is this some kind of error message? The script worked fine before (and no, it's not a script that makes led 13 blink in case you wonder).
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Upload an example sketch ( blink) and see if works with external power supply , then you will know where the problem lies.
What is your external power source is it upto the job ?
In my experience
The fading indicates that the boot loader is waiting for the data to be uploaded; this state will last a few seconds. That should not happen after a power cycle; after a power cycle, the boot loader immediately passes control to the user program.