I want to register an app that will use the V2.0 API here: https://stackapps.com/apps/oauth/register. However, the app will run on a mobile phone so I cannot enter a domain name.
What can I possibly do? Enter a fake domain name?
You can register apps that aren't web apps.
This page: http://api.stackexchange.com/docs/authentication, gives details for registering a Desktop Application. A mobile phone app comes under the category of a desktop application.
You should use the implicit flow, so check the box at the bottom of registration form, and for the OAuth domain use:
stackexchange.com/oauth/login_success (note the lack of https://).
This is also the value you should use for the redirect_uri parameter (although with the https://). Then in your app you need to check for when the web browser loads stackexchange.com/oauth/login_success page.
It probably should be clearer on the registration page.