There is no "option of a trial account" when you have bought a licensed version of Lr.
As to why Lr won't start, you have to give more details:
Are you getting any error messages? What do they say? Please quote any meessages with exact text shown in message.
BTW: I'm not a Mac person, I know Windows only, so somebody else on this forum has to give you advice.
Since you say that you haven`t found a < .lock> file, it`s most probably not a Lr problem but a Mac problem.