There are multiple factors that cause the corner of the mouth to turn down. First of all, it often runs in families. It can also be affected by thinning of the mandible bone. In addition, in this area all the muscles of the corners of the mouth come together and can pull down as you age. The correction is often involves more than one product injected into the area. A small touch of botulinum toxin (Botox, Dysport, Xeomin) can release the muscles that pull down. The depression can be filled deeply with calcium hydroxyl apetite (Radiesse) used to fill under the corner and into the marionette creases. On top of this I like to layer a hyaluronic acid like Belotero Balance.