  Integrated and differential $J/\psi\ R_{AA}$ are systematically studied in
Au-Au collisions at RHIC-BES energies in a transport approach, including cold
and hot nuclear effects respectively in the initial condition and the collision
terms. With decreasing energy, while the temperature, life time and size of the
QGP fireball decrease, the nuclear absorption of the initially produced
charmonia is more and more strong, and the nuclear shadowing effect on
charmonium regeneration goes to anti-shadowing first and then to shadowing
again. As a competition between the cold and hot nuclear effects, the QGP phase
is still important for charmonium production at $\sqrt{s_{NN}}$=200, 62.4, 54.4
and 39 GeV but becomes negligible at $\sqrt{s_{NN}}$=14.5 GeV.
