Lip fillers - upturned corners.

Lip fillers can be used to define the border of the lips. I would recommend asking for a hyaluronic acid based filler such as juvederm to create natural results that can easily be dissolved if you do not like the results. This type of filler will be temporary so it gives you the opportunity to assess whether you feel fillers are the best option for the look you're trying to achieve. Botox can also be used to create the appearance of the upturned corners. 

I would use Juvederm Volbella lip filler. A special technology called Vycross is used which allows the filler to be injected in single strands which holds less water and gives smoother results. Since you are concerned with the upturned corners more so than getting plumper lips, this is a great way to specifically target that area and give you the subtle results you are looking for. Best, Dr. Emer. 

Thank you for sharing your excellent question.  Lip fillers can be used to turn the corners of the mouth upward, and as you are interested in creating greater definition than volume I would advocate for a thinner hyaluronic acid dermal filler such as Volbella.  Hope this helps.