Since I don't know the distribution of lesions (I assume only around the mouth the way this is worded),I will address this question in a general way as if distribution did not matter. There is usually an immune reason for this kind of skin problem, either a hyper response, such as an allergy, auto-immune disease, scabies, etc.; or a hypo response, such as immune system impairment/failure, demidicosis, bone marrow stem cell failure, cancer, etc.
Response to antibiotics does not give a diagnosis, merely that the problem is antibiotic responsive. You are treating a symptom and not a root cause.
My suggestion is when the next episode happens, then a full blood panel (complete blood count, biochem profile, thyroid, and immunoglobulin levels), plus or minus a bone marrow aspirate. Skin scrapings to rule out ecto parasites, such as Demodex and Sarcoptes; hypo allergenic food trial for up to six months, and abdominal ultrasound exam may be required to categorize this problem.
I know it sounds incredibly expensive, but these 8 week trials with antibiotics aren't working vey well, and are also expensive. Talk to your vet, start running the simple basic tests and if need be, then work up to the more extensive tests.
But find a cause for this problem, don't mindlessly throw pills at it. Good luck.
Dr. Van Lienden
Dr. Raymond Van Lienden DVM
The Animal Clinic of Clifton
12702 Chapel Road, Clifton
Virginia, U.S.A. 20124