Healthy weight is not a matter of how much you eat so much as what you eat.
Make sure you're feeding the whole family healthy meals, and start sneaking ground nuts, seeds, avocados and, easiest of all, just some olive oil, into your underweight son's meals. So, just before you serve the soup, drizzle a tablespoon or so of olive oil into his bowl. Same for pasta or whatever. That way your overweight son doesn't notice that his brother is getting something that he, himself, is not.
Make sure that whatever snacks are available to your kids are healthy ones. Don't think in terms of healthy foods/snacks and unhealthy treats. Make even the treats healthy. This isn't as boring as it sounds. Just start switching out. Instead of regular chips, start buying pretzels. Instead of ice cream, get So Delicious (4 grams of fat, I think, per serving instead of 11 or 17 or whatever). Instead of sugary soda, start buying E-mergen-C vitamin C drinks (just one per day - kids' version) and fruit juices, and adding to seltzer water or club soda. As for meals, start replacing your fattiest meat meal with, say, a pasta salad or something with no meat in it (even if you want your younger son to have more fat, fat from meat isn't good for anyone). Etc. It will take some adjusting on everyone's part, but it only takes a few weeks to change a habit.
Also, maybe the underweight son is more active than your overweight son? Get your 11-year old involved in some sport, a martial art, or other activity that keeps him moving. Don't be afraid of getting your younger son involved in active play, as well. This can build his appetite.