Cleaning stove top

My (reasonably new) gas range is all black with sealed burners. I've been having a heck of a time trying to clean around the burners -- it's impossible to get off the burned-on spills which get set as soon as they occur, and it almost seems as if the finish is "bubbling". I can't think of a solvent I haven't tried, and I've been tempted to get out my scrubber, but am afraid it will scratch the surface and make it worse -- but at this point, I can't imagine how much worse it can get.

Any suggestions you might have would be greatly appreciated.