Can a father get custody of the 3 children in Louisiana when he has very sold proof that the mother committed adultery?

This is the 21st Century.

Adultery has nothing to do with whether a cuckolded husband can get child custody, and it hasn't for a very long time.

Can a father get custody of the 3 children in Louisiana when he has very sold proof that the mother committed adultery?

A woman having an affair has nothing to do with being an unfit mother. If that is what you mean.

Which parent gets custody is generally decided by negotiation or according to the best interests of the child(ren). The adulterous parent is generally not in any worse position in these times.

Sorry no, having sex outside of marriage will not effect custody at all. I have seen cases where hookers ( women who have sex for money) still get custody of their children since they take good care of them.