In criminal law what constitutes child abandonment?

In the state of Ohio, can the biological mother/custodial parent be prosecuted for dropping off an infant with the child's biological father or paternal grandmother? Will that situation be grounds for a complaint with child protective services? The biological father is unwilling to ist in caring for the child and the mother has to return to work, but cannot currently afford child care and is reluctant to apply for public istance. The infant is 2 weeks old.