What is the best solution to get rid of the fat lumps under my eyes? (photos)

Fat lumps can occur after transfer to the lower lids.  The skin is very thin and the fat can be seen.  You can try a couple of things.  Ulthera can help to shrink the fat - may need 2-3 treatments.  You can also try some steroid injections to shrink the fat.  If these don't work they can be removed surgically.    

I do a good number of fat transfers and actually the lower eye / orbital rim  (upside down C area) is one of my favorite areas for fat transfer. I have had similar (albeit smaller) lumps over the years and have found them to almost always respond to very diluted injections of triamcinolone. Low concentrations of the steroid are important and waiting about 6-8 weeks between injections has worked well for me in these related situations. Best of luck to you.

Sorry to hear that you have this problem and it has persisted for so long.  The fact is lower lid fat grafting can be very challenging, as the take of the grafts can be unpredictable and the thin skin can reveal superficial contour irregularities.  It must be difficult, but you should know that there are approaches that may be able to produce significant improvement.  The fat deposits may respond to injections of steroids (triamcinolone) or an antimetabolite, 5 FU (fluorouracil).  Aspiration of the fat with small bore cannulae has been described, as well. If these minimally invasive approached prove insufficient, an attempt could be made to surgically excise the fatty deposits.  Good luck.