Can my ex sue me for child support and custody?

I have a 15 year old son who is saying he wants to go live with his dad because I’m too hard on him. His dad lets him get away with whatever he wants and I’m concerned my son won’t keep up with his homework or go to his school tutor if he lives with his dad. My ex is saying that if our son does go to live with him, he’s going to sue me for the same amount of child support he’s been paying to me. Can he sue me for custody of my son and child support just because my son doesn’t want to follow the rules at my house?