The question is suited to FAQ 211 ("Why am I having trouble logging into LiveJournal?"), but the answer can be found in FAQ 162 (e.g., cookies.)
I'm thinking of editing FAQ211 to point to FAQ 162 to make this case easier to handle. That could be its own bullet point, or be tacked onto the end of the "Change Browser Security Settings". Hm, possible wording:
If you received no error messages when logging into LiveJournal, but find that you are unable to log in, your browser's security settings for livejournal.com may be may be preventing your login cookie from being set. You should name LiveJournal as a trusted site or manually lower the browser security settings for livejournal.com specifically. Consult your browser documentation for instructions. [[somehow, pointer to FAQ162]