are there any problems with bishul akum by a micr-wave?
Anything involving an akum is a potential problem.
A microwave is not usually used to cook food from raw, but for reheating, in which case there is no problem.
In any case, microwaved food is not oleh al shulchan melachim, as any chef will tell you. On the other hand, some poskim hold that if the general type of food is oleh al shulchan melachim than it makes no difference if this particular preparation of it is not. I believe there is a relevance here to the debate about chips/fries/roasted/cooked potatoes.
Many microwaves have a combination of mirowaving and also cooking. U gotta know the facts.
If you leave food in the microwave for longer, it gets cooked.