You can have surgery to correct this drooping of the lid which is usually since birth called Ptosis. The cure will be permanent after surgery but it is possible that it may be over or under corrected.
If it is not since birth, you must rule out other causes of ptosis like Mysthenia Gravis which may be treated by medication.
Surgery should be done by an oculoplastic surgeon and any big eye hospital or institution will be able to do it.
Cost will vary from surgeon to surgeon.