How to run desktop application with database locally without installing any database Server?

You can use SQL Server Compact, which can either be installed separately or deployed as part of your application.

See here: How to deploy SQL Server Compact Edition 4.0?

And here: https://msdn.microsoft.com/en-gb/library/aa983326(v=vs.110).aspx

Your DB code in your app - apart from your connection string - should not need to change as you're already using some version of SQL Server.

There is also SQLite, which is as far as I know is well regarded, but I know very little about it. http://www.sqlite.org/

Hope that helps!