Sole Custody of my child?

My son is now 6 years old, and his father has had little to do with him since the day he was born. In fact, as he was not present, and I was unable to locate him at the time of birth, he is not even listed on the birth certificate. They did not meet until this past November, when my son was 5. At that time we did a DNA test, and they met a few time here and there. I have yet to receive a dime in child support. While I was hopeful for a while, I am beginning to realize that he is not interested in a relationship with his son, and has never offered us any financial support. While the support would definitely make our lives easier, and I of course would rather there be a father-son relationship than nothing at all, I have lost any faith in the situation. What I want to know is, am I required to apply for legal sole custody, so that the "father" can't show up years down the road show up and claim parental rights when it suits him? Or is the fact that no where is he listed as the father sufficient? Do I have legal sole custody no matter what?