The phone says K790
a...which means it is used for North America. SE should not have a problem registering the phone's
IMEI number because it was made for use in NA.
Just remember, if you wish to use the warranty through SE, you'd have to ship it to the service center. You also might have to wait for an uncertain amount of time. Usually you'll be informed that it would take perhaps 2 weeks. But some things might occur, and you could be out of a phone for maybe a month (or more). But with this way, you have less of a hassle.
If you want to try it out on your own, you can use these steps. First, do the basic troubleshooting. Master reset, taking out the battery, blah. If the problems still persist, try updating the firmware through the
Sony
Ericsson
Update
Service on your cell phone. If that doesn't work, try flashing it through services like wotanserver or davinci team (this way is more of a hassle, but you don't have to worry about if you're actually going to have your phone returned in a timely manner or not). Google search them, and you'll get those links to the sites.