There was an issue for some people with logging into Lightroom mobile on the Mac last week. Typically the error message from Lightroom Desktop was :
There is now a solution to this problem,
This is a known issue. The following workaround will allow you sign-in access.
1. Open System Preferences in Mac OS.
2. Select Sharing.
3. Enter your computer name into the Computer Name field. The computer name you need to enter into this field is in the text directly under this field, and contains a .local suffix. The computer name you need to enter is this name, but without the .local.
Note: If the computer name in the text is just .local, enter whatever computer name you wish into the Computer Name field, but do not include the .local.
4. Close System Preferences.
Information was originally posted here.