Botox should never be used in the lower face except by an expert injector because it can cause facial drooping (like a stroke only temporary) or problems with chewing and drooling. I will sometimes put a tiny drop of it in the muscle that pulls the corners of the mouth down ( the depressor anguli ori).

Fillers are a much better solution for this. Generally Juvederm/Juvederm Ultra Plus or Restylane/Perlane are used.  Cost depends on your age and how much loss of support you have around your mouth. One to two syringes would be the average.  Dr. B