The Pokémon Go Battle League is the PvP part of the mobile augmented reality game. In most cases, the PvP service is typically active and you can fight managers from all over the world if you have tickets to fight them. The first thing you want to check is your internet service, and hopefully the next step is to see what's going on behind closed doors and find out what problems the Pokémon Go application might be having.
An excellent first step is to investigate the Pokémon Go blog to see if they have any updates on the service. For example, you may experience little hiccups in the middle of the season, such as a portion of the service that goes out for a while or changes the release date for one of the four leagues. The blog on the Pokémon Go page does a great job of breaking down what's happening in the game's current events and explaining how they plan to adjust for any changes to the schedule.
After that, your next step is to jump on Twitter and check out Niantic's Twitter Account Support. The people over there are extremely good at informing the community of any problems that occur with any of Niantic's games, but the ones that run it are exceptionally good at informing the community of Pokémon Go-related problems. If you don't follow this Twitter page, at least bookmark it on your smartphone browser to quickly jump to it to see if there are any recent posts on the declining Battle League and what is being done about it. This statement applies to both planned and unexpected maintenance.
These are the two main ways that you will have to rely on to know Pokémon Go Battle League not available. Outside of these two options, you must mainly try restarting your Pokémon Go application, checking your internet service and making sure you have a strong signal. If you are experiencing performance issues when trying to connect to the Battle League and are unable to log into it anymore, you'd better wait to fight with other trainers again until you return to a location with a reliable WiFi connection.