Certainly, your characters - whether they are superheroes are not - should have foibles. They should have problems; they should have things that their powers can't solve. That's what makes them nuanced, interesting characters. They can have intense motivations. They should have intense motivations to do what they do.
If you go back and look at the first issue of 'Indestructible Hulk,' if you have a sharp eye, you'll catch something that I totally forgot to put in there. In my horror, I only realized after the fact that I took totally for granted that everyone in the world knows what triggers the transformation.