The welcome screen opens but it then says that the trial period has expired and will not go any further. I should say this is a legal, fully licensed copy that has worked just fine for the previous 9 months.
I tried uninstalling and reinstalling but it made no difference. I tried lots of suggestions from the Adobe forums, but this is the one that actually worked.
You need to rename 2 folders, SLStore and SLCache, to SLStore.old and SLCache.old respectively.
C:\Program Files\Common Files\Adobe
(ProgramData is a hidden folder so remember to 'show hidden folders'.)
C:\Program Files\Adobe\Photoshop Elements 12\PhotoshopElementsEditor.exe
right click and 'Run as Administrator'.
The program should now open. You will have to sign in with your Adobe ID and provide the licence number to reactivate the licence. You can get your licence number from your Adobe account online, if you have lost it.
From now on the desktop shortcut or any other shortcut should work as normal without having to 'Run as Administrator'.