the link has formal procedures: https://***.bamf.de/EN/Themen/AsylFluechtlingsschutz/Sonderverfahren/Flughafenverfahren/flughafenverfahren-node.html
Just so you know, if you apply for asylum at the airport, they'll take your passport, and you'll get a quick decision from the court, maybe within a week or less. But if your life isn't really in danger and you're from a safe country, things get tough because you came here on a visa. People whose lives are truly at risk don't have time for visa's and all that paperwork. Plus, if your case doesn't make sense or seems weak, as a minor, you can stay here until you're 18. But once you hit 18, you're immediately sent back home.