Hi @CHRISTOS STAVRINOU ,
...How to bypass this limit. ...
You can't by pass this limit. For one instance, if you exceed the maximum number of user connections, you receive an error message and are not able to connect until another connection becomes available.
...What is an alternative solution to connect more users.
However, if you have more connections,one solution is you can install more instances in the server. SQL Server supports 32,767 connections per instance. You can run 50 instances per node, so the maximum number of client connections per server node is 32767*50. (Note: please check the version of your sqlserver firstly, max number per server node is 32767 * Max Instance Numbers)
Please reference: configure-the-user-connections-server-configuration-option, maximum-capacity-specifications-for-sql-server
And another solution is the reply from this case: connection-limit-of-32767
Another solution is to use multiple SQL Servers, arranged in some order to limit connections (Department, Location, etc), and MERGE replication to keep all copies up to date.
BR,
Mia
If the answer is helpful, please click "Accept Answer" and upvote it.