I had lip injections 2 weeks ago and developed cracks in the corners of my mouth at injection site.

The injections may have irritated this area or could have be caused from consistent dampness post injection. Consider a hydrocortisone cream bought over the counter. If the hydrocortisone cream does not clear the area in 10 days schedule a follow up with your injector and express your concerns.  

Yes, that is angular chelitis, and can occur in anyone, recent filler injection or not.  Agree with Dr. Rueckl's treatment suggestions, and another option is to put over-the-counter miconazole or clotrimazole anti-yeast cream (sold as vaginal or foot yeast treatment) with over-the-counter hydrocortisone cream, two x/day for a few days.  Some people get this from saliva getting trapped in the corner of their mouth while sleeping esp. if they wear a mouth guard or have dental gear.  Others get it from excessive licking.  Sometimes we don't figure out exactly what triggered it. 

Sometimes this happens. Vusion is a prescription that is really great, but you'll need to get an Rx from a doctor for it. If you don't want to do that, try getting Dr. Dan's Cortibalm online. It's a chapstick that has hydrocortisone cream in it so it helps heal the lips rather than just moisturize them. You can buy it on Amazon, or you can go to a pharmacy and they can order it for you over-the-counter. It's only a few dollars, but it's great! "This answer has been solicited without seeing this patient and
cannot be held as true medical advice, but only opinion. Seek in-person
treatment with a trained medical professional for appropriate care."