Dark discoloration under the eyes is often from blood vessels that are close to the skin and cause the dark circles. Allergies, lack of sleep, and dehydration are some factors that will exacerbate the dark circles.There are cosmetic products that can help. I am not aware of any cosmetic procedure that will eliminate them. I have used a product by DHC, a Japanese cosmetic company, that is effective in fading the dark circles. The product is called Eye Bright. It is much less expensive than Hylexin and it worked better than Hylexin for me. You may want to try it.
I, too, have had dark circles under my eyes since I was a child. I have used tons of products to try and hide them or fade them. Eye Bright has been the best. Over time the circles will be greatly reduced and faded. You can see a difference in just a day or two of use but the effects are greatest with continued use. I spoke with a cosmetologist at the company and she told me the product has caffeine in it and that is the active ingredient. It makes sense as caffeine will vasoconstrict blood vessels, shrinking them thus they are less visible. You may be able to request samples from the company.