You really need to inform yourself of the consequences of what you want to do. They cannot pay your mortgage or if they do, they are gaining an interest on the house and that could have some repercussions if things were to go wrong.
If they pay you rent, you become a landlord and with that, your arrangements with them becomes a business.
Of course, your sons are reasonable, love you and all and you can't see anything going wrong.... yet it can so easily be so. Say your sons get girlfriends, and they move in. They become sub-tenants. Things go wrong between them, the girlfriend(s) refuses to move out. Your sons can't stand living with them any longer and move out.
In the meantime, your business fail, you end up without a penny to your name, you want to move back in so that you get at least a roof over your head.... except that by then, your house is inhabited by two strangers, who are refusing to move out and don't pay rent. Your sons can't afford to pay rent where they are and to you. You need to go through the process of eviction, which can take many months, even worse if you haven't followed all the proper rules excepted of a LL. Before you know it, you have to employ experts, and it's all costing you a fortune through not fault of yours or your sons.
It doesn't always happen to others.