I think that if you truly "gave up", you wouldn't be posting in this forum. I smite thee!
I was sympathetic at first, but now I'm wondering if you only want to sign over your parental rights because you don't want to pay child support?
I am a step-mother of many, and my DH's oldest daughters 22 and 20 do not really have a relationship with him... well the 22 year old comes around mostly due to her younger sisters (5 and 8 years old). 18 years ago, my DH had been attacked by a group of the 1st BM's brother and family with brass knuckles and even was shot at, because BM was vindictive and wanted her new "man" to assume the fatherhood role to their daughters. It's ironic, because she is no longer with him and "step-dad of the year" doesn't even claim my oldest SD's as he now has a "new family".
I say that to say this: My DH regrets every day the decision to not fight for his oldest daughters. Even with the two youngers ones, as a stepmom to them, I worry and wonder, what happens if we will get them full custody? I'm pretty sure that their behavioral problems that they are showcasing with their mother will still linger. I worry about having to take them to see a psychologist, I worry that sometimes when they aren't being "not-so-great kids" did I sign up for the wrong class and will I flunk it because, I too, as a rational adult, am responsible for them. Would it be alright when they do something sub-par to say "well, that's what they DO when their with their mom." or "well, she's like that because of the environment that she lives in." I think no. I think it's not enough to excuse their behavior. I tell them all the time, I'm sorry you were dealt such a bad hand. That may be the way you're allowed to act/allowed to fail tests/allowed to not turn in homework/allowed to not clean when you're elsewhere with another adult, but you will not act that way when you are with me. It's hard and it's not easy, but just like when I married my DH (and even before) I signed up to be a part of the parenthood.
You can't revoke your status just because she, as an 11 year old has problems. That should make you want to save her more! Don't be the guy, awkward at family gatherings 10 years from now trying to claim her.
That would be sad.