Try to add fertilizer to the soil or you can use an artificial grass at the corner. They look same as real grass. I also had a same problem. So I installed Village Green Turf in my lawn. Now my lawn stay green in all seasons.

What kind of grass do you have (or want) growing there?

The problem with dog urine is the soil does not have sufficient population of microbes to process the urine. One thing you might try is putting some shredded mulch, a little organic fertilizer (any flour or ground grain from the kitchen will do), a handful of table sugar, and keep it moist for a few weeks. That will do wonders for your microbe population and should help with the urine problems. The microbes will eat the sugar and wheat/corn and repopulate the soil. If it starts to work and doesn't you get a setback, do it again. You can apply ground grains as often and as much as you want.

If you have fescue you will need to re seed or resod. Need to know what grass you have. Pictures are always nice.