You can't write to SQLite from multiple processes either, so you mitigate that exactly the same way you do for a desktop app (i.e. show "Can't access DB for writing because other instance is running")
You can write from multiple processes and this is enabled by default (see PRAGMA locking_mode [1]). If you've got multiple writers you might have to occasionally handle SQLITE_BUSY and retry.
[1] https://www.sqlite.org/pragma.html#pragma_locking_mode