Yes, I have had exactly the same thing for about a year now. At times, working with weights, I have been unable to shift even 5kg with the knee in question, while shifting 40kg with the other leg so it definitely is weak.
I have no pain, just a vague sense that the knee "isn't right." Very much what you describe. It has never locked up or given way while running, although it is sometimes an effort going up stairs.
I have been to my GP and she did what I assume are the standard range-of-motion tests without finding any obvious problem. Her advice was to build the knee up in the gym, and go to a physiotherapist if that didn't work and I still felt there was a problem. I've done the former (it hasn't made a difference) but not the latter (can't afford it now).
So, like you, all advice welcome.