I had the same problem. options I could think of was train the dog to go in
a specific spot. tie dog up so it cant roaming into areas I did not want him
to go, or just patch the spots that die with grass seed and some elbow
grease. I did the patching of the lawn and it worked out well.

I did see a article on the TV about a product you would feed the dog on a
daily basis that would prevent (kinda like a daily vitamin) the lawn from
dying. Not sure if it actually worked or not of if it is still available but
it may be worth a call to the vet and ask.