One of my dogs was suffering from this during the year. He had several visits to the vet, several cycles of a rather expensive antibiotic, had to follow schedule of washing the folds at the corners of his mouth several times a day....and the infection would always come back. (red, inflamed skin in the folds at the lower corners outside the dogs mouth).
The vet said that some dogs with a deep lip fold have to have surgery to smooth out the fold and prevent the moisture that fosters the infection, or they just have to constantly be on an antibiotic.
Um...not anxious to do either one of those choices.
Did some research online, and found that German Shepherds often seem to suffer from this lip fold pyoderma. A GSD forum talked about using a cream called Penaten cream. Several of the people had good results. I found it on Amazon for about $8, and thought what the heck...seems to be mostly zinc oxide and some herbs, probably can't hurt. People use it on babies for diaper rash, its so gentle. I believe it is only sold overseas, but you can order it on Amazon.
Applied it 3 times a day (it comes in a can, and a little goes a very, very long way, will last forever). Within 1 week, the active infection was gone. He can't really lick it off, since the infection is on the outside skin at the corner folds of his mouth.
Now I apply it about twice a week to keep the pyoderma away, or whenever I see the skin start to get the slightest pink.
He hasn't had any infection at all now for several months. Thought I would mention it in case anyone had a dog or cat who has the same thing. Worth a try if antibiotics don't help.