One that I've never seen mentioned before and that I've personally used is HESK.
Lightweight, PHP front-end with SQL backend. Fairly easy to install if you have a little knowledge on how to work around PHP.
We're a small company so we use HESK. It's easy to have several helpdesks because of the copy/paste nature of it and we work in nearly three-dozen schools and have a helpdesk for each to stop there being any confusion over which helpdesk is which. Each helpdesk is on a different website so we don't get confused either!
We use HESK for our helpdesk. I wasn't involved in setting it up but we have a helpdesk per school we work at which automatically emails selected addresses when a ticket is created/update. Requires credentials to access, can close/open tickets, customer receives an email when replies are given, custom fields can be created, etc. Everything you really need from a basic helpdesk and, as far as I understand it, it's just a matter of hosting it on a new domain, pasting in the basic info like Wordpress and then configuring accordingly.
That's a very basic and perhaps messy explanation but HESK is what we use and it seems to work well enough.