This is a common concern! A little bit of botox or dysport into this area can fix this wrinkling. It is usually due to an overactive muscle. Please consult with a board certified specialist in the face who can help and guide you towards achieving the results you seek.
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The problem here is maxillary (upper jaw) over growth. The surgeon would need to surgically reposition your upper jaw. This will assist you in closing your lips without any muscle overuse. The cost would be some where around $20,000. This includes orthodontic treatment as well.
Chin implant vs orthognathic surgery
From the cosmetic standpoint a chin implant would work very well for you although it does look like you have some malocclusion. In that sense, othognathic surgery would be a good option, though it involves far more surgery and much greater expense. You would have to consult with a surgeon who does these procedures since cost can vary greatly. The chin implant is a relatively simple procedure and would be relatively inexpensive. It would likely be my first choice.
You have an overactive muscle in this area. A little botox or dysport will help to relax the muscle and should decrease the wrinkles. Will need to repeat every 3-4 months.