They are often praised by adults for their respectfulness and good manners, but have always been competitive with each other and are getting worse. I do not know how to handle this or what type of discipline, if any, would make them change. Both have been raised with good values and attend private Christian school. I know they love each other, but just won't admit it. I think they are trying to get attention to establish who's the boss. When confronted about their actions, the younger one denies any wrongdoing, and the older one breaks down and accuses me of always blaming him for the fights.
I am at my wit's end. Their dad is no help. What do I do?
Generally, parents should stay out of children's arguments, but parents should, with the children, establish the rules of arguing. The first and basic rule is no hitting. You can be very creative with the rest of the rules. For example, predetermine a room where arguments are to be settled and a consequence for both sons if they don't settle. Then, if arguing gets too loud, calmly tell the boys you know they can figure out a solution and that they are to go to the room, settle the argument and come back and tell you what they have decided. They will probably try to draw you into it, but stay out of it.
Be consistent and if after a reasonable time they haven't settled, impose the consequence on both. For more help, I suggest looking into any parenting classes that might be offered in your area. These classes are great for getting support and ideas from other parents going through the same trials of raising an adolescent.