The best solution is to reserve an area in your yard that can be covered with non-plant material such as mulch or gravel. It is best to select a location that is distanced from potential smell. Train the dogs to use this non-turf area.

Dog urine is a salt, which affects plant growth. Repeated urination on anything that is intolerant of high salt levels will harm or kill the plant which is why there is no remaining grass.

Water is the only thing that can reverse the effects of dog urine. Some may suggest using baking soda, dish washing detergent, or products claiming to dissolve urine and odors. These are not effective and can actually compound the problem.

To revive the lawn, it may be necessary to reseed or resod. Heavily water the damaged spots for several days before reseeding or resodding to leach thru the salts.

Additional information of dog urine and home lawns:

http://www.ext.colostate.edu/mg/gardennotes/553.html