We had the same problem with our 3 year old lab mix dog. First we changed her food to eliminate corn and wheat from her diet. The Vet prescribed a 2% lime sulfur dip that we used once a week for a month.
During lime/sulfur treatment she was on 500mg of Cephalexin 2 times a day. After the lime/sulfur treatment ended we bath her in Virbac
Etiderm shampoo once a week. We also had allergy test run and found
she was allergic to common things found in a normal house. She receives
a monthly allergy shot for those allergies. Her coat cleared up and has been clear now for 6 months. Good luck with resolving you pets issues.
This can be a costly and time consuming process.