If the steroid injections are
effectiveEffective strength cough syrup than the problem could be allergies or autoimmune disease.
A biopsy may be a good idea, but if you are hesitant about the idea, the following options could be tried:
Skin scraping to check for demodex, scabies, or other mites.
Culture and sensitivity to check for bacterial or fungal pyoderma
Fungal Assay
Hypoallergenic food trial
Try the following medications: Atopica, antihistamines, and oral steroids, antibiotics and antifungals for the secondary infections.
Allergy testing (skin testing or blood test)
Desensitization