I painted my kitchen cabinets. Worked well but over time they have become sticky in areas often touched; near knobs. I think it is a bit of grease from cooking but when I try to clean them, paint comes off and they are still sticky. The rest of cabinets are fine.
If the paint is peeling off the wood surface then most likely there was a problem with how the cabinet was prepared for the paint or with the paint itself.
Painting any surface located in a kitchen takes more effort and attention to details when prepping it for paint. As you already know grease from cooking that becomes airborne in water vapor will adhere to wall surfaces and accumulate over time. For proper paint adhesion it must be removed through scrubbing with a de-greaser-type cleaner. Most commonly used is the cleaner labeled or containing "TSP" (Tri-Sodium Phosphate). Added to warm water and applied with a sponge (or sponge mop) the cleaner easily cuts through greasy build-up. Rinse the applicator often and then re-wipe with fresh clean water to remove any residue.
Since the cabinets are already peeling it will be hard to hide the edge of the paint with a new coat. You can try sanding the edge to "feather" or angle the transition so when another coat is applied it may be less noticeable. But I would guess since the wood hadn't been prepared you will continue to have peeling issues. It may be wise to start anew by sanding all the door faces to remove new paint, clean with TSP and prime and finish coat.
The paint isn't the main issue here. Repainting is just a temporary fix as the grease comes back. If it's upper cabinets as well, then your exhaust fan isn't working well enough and needs to be cleaned or replaced. However, I'd suggest just looking up some good degreasers to clean it off with. Dawn, magic erasers (could harm the paint), etc.