If this could be Seborrheic Dermatitis or Candida, a yeast supplement called Saccharomyces Boulardi is probably best to treat it. In addition to ingesting the pill, I would open it and mix it with water to make paste for lips.
"S. boulardii and its extract significantly inhibited C. albicans adhesion to epithelial cell lines. Our results indicate that S. boulardii affects C. albicans adhesion and reduces cytokine-mediated inflammatory host response."
The following web page also lists these probiotics to cure dermatitis: