Dark patches around the corner of the lips can be caused by many reasons such as hyper-pigmentation, hormonal imbalance and multiple other factors. There could be multiple factors involved but in ur case there could b certain medicines that can cause pigmentation, which may also affect the area around the mouth. These include tricyclic antidepressants. It can be treated with medication and treatments I'd required. Consult a Dermatologist for the same.
Health Tips
keep the skin around the mouth well hydrated. avoid cosmetics