I did some research on this topic for you.
I have found that there are three causes for dark circles under the eyes: too much pigment, too many blood vessels, or a shadowing effect.
If you have excess pigmentation in excess, lasers for dipigmentation or chemical peels can help. .
Lasers can help if you have an excess of blood vessels.
People with with deep set eyes will have the shadowing effect.
You can also use Restylane or Juvederm. Restylane does not last as long, and from personal experience I prefer Juvederm. They claim the injections last 6 months, but most last 4 on average. I've had Restylane last for only three months (not favorable). For under eye circles, these injections must be placed deeply under the muscle. If you do not inject hyaluronic acid deep enough you will get a black and blue type bruise.
Sculptra and Radiesse can last for up to 2 years or more, but usually about 18 months on average. These are not necessarily for under eye injections.
Hope this helps!!!