Can you get botox injections around your mouth?

I don't think so.

The best option would be Hyaluronic acid (I read about it somewhere).

Besides, Botox interferes with the ability of the muscle to contract normally, and dealing with your mouth... hmm, paralyzing that crucial muscle of yours, even for a bit, doesn't sound like a good idea. In any case, consider interrogating your doctor about issues like that.

Visit specialized websites about surgeries and other medical procedures if you want to know more about things like that! I have a good suggestion: http://www.surgery.com

wish you the best!