This is a rash around the mouth caused by excessive licking or thumb-sucking. The lips are also dry and sore. The saliva irritates the lips and skin. Lip-licking and thumb-sucking are habits. The red scaly rash around the mouth, and the dry, chapped lips, get better when the habit stops.
1% hydrocortisone cream for a few days clears the rash. Petroleum jelly (Vaseline) can then be used to protect the skin. Lip cream will moisturize and protect the lips.