  We study the isotope effects arising from the coupling of correlated
electrons with dispersionless phonons by considering the Hubbard-Holstein model
at half-filling within the dynamical mean-field theory. In particular we
calculate the isotope effects on the quasi-particle spectral weight $Z$, the
renormalized phonon frequency, and the static charge and spin susceptibilities.
In the weakly correlated regime $U/t \lesssim 1.5$, where $U$ is the Hubbard
repulsion and $t$ is the bare electron half-bandwidth, the physical properties
are qualitatively similar to those characterizing the Holstein model in the
absence of Coulomb repulsion, where the bipolaronic binding takes place at
large electron-phonon coupling, and it reflects in divergent isotope responses.
On the contrary in the strongly correlated regime $U/t \gtrsim 1.5$, where the
bipolaronic metal-insulator transition becomes of first order, the isotope
effects are bounded, suggesting that the first order transition is likely
driven by an electronic mechanism, rather then by a lattice instability. These
results point out how the isotope responses are extremely sensitive to phase
boundaries and they may be used to characterize the competition between the
electron-phonon coupling and the Hubbard repulsion.
