Thank you for your question and photograph. Botox cosmetic can be injected to elevate the downward corners of the mouth. My advice would be to consider a dermal filler to augment the depressed areas in the marionette lines. Best of luck.
Botox is more commonly known to treat the lines in forehead and around the eyes. However, it can be used to relax the muscles in the lower face as well. In my practice, I often use Botox to relax and soften the downward turn of the corners of the mouth. Based on the photos you provided, I believe you are a better candidate for volume replacement with dermal fillers to help soften the crease of your marionette lines and soften the fold of your nasolabial folds.