Did you read this? You can apply for a passport based upon the child's birth certificate if no father is identified on the birth certificate.
Quote Quoting One Parent (with sole legal custody) MUST:
* Appear in person with the minor

* Sign Form DS-11 in front of an Acceptance Agent

* Submit primary evidence of sole authority to apply for the child with one of the following:
* Minor's certified U.S. or foreign birth certificate listing only the applying parent

* Consular Report of Birth Abroad (Form FS-240) or Certification of Birth Abroad (Form DS-1350) listing only the applying parent

* Court order granting sole custody to the applying parent (unless child's travel is restricted by that order)

* Adoption decree (if applying parents is sole adopting parent)

* Court order specifically permitting applying parent's or guardian's travel with the child

* Judicial declaration of incompetence of non-applying parent

* Death certificate of non-applying parent
NOTE: If none of the above documentation is available, the applying parent must submit Form DS-3053 stating why the non-applying parent/guardian's consent cannot be obtained.