OK. Back to my original problem. It was a problem with code signing the gdb library. I renewed code signing and debugger and app are starting.
So there remains one thing: After having started my mac and then using Lazarus IDE and debug mode, I have to enter my admin account data. As long as i don't shutdown my mac, no further need for that. It is not a matter of Lazarus. I can start and quit Lazarus; if I don't shutdown and restart the mac, no further input of admin account data. So it is not really enerving, more a question of interest.