I recently upgraded MariaDB from version 10.5 to 11.4 for a couple of RHEL9 servers. Here are the steps I used:

Add the MariaDB repo containing version 11.4 and exclude MariaDB 10.5 from the system default AppStream repository.

Uninstall MariaDB 10.5 and install version 11.4:

$ dnf remove mariadb
$ dnf install mariadb-server

Create socket:

$ mkdir /run/mariadb
$ chown mysql:mysql /run/mariadb

Start and enable MariaDB:

$ systemctl start mariadb
$ systemctl enable mariadb

Run the upgrade script:

$ mariadb-upgrade

Check if selinux is blocking mysql requests (it was in my case):

$ journalctl -t setroubleshoot

Make sure socket is created on reboot (overrides that in /usr/lib/tmpfiles.d/mariadb.conf):

$ vim etc/tmpfiles.d/mariadb.conf
# Type  Path         Mode  UID   GID   Age Argument
d       /run/mariadb 0755  mysql mysql -

Check status and logs:

$ systemctl status mariadb
$ vim /var/log/mariadb/mariadb.log