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Using the large-$N$ limit of the $t$-$J$ model and allowing also for phononsand the electron-phonon interaction we study the isotope effect $\alpha$ forcoupling constants appropriate for YBCO. We find that $\alpha$ has a minimum atoptimal doping and increases strongly (slightly) towards the underdoped(overdoped) region. Using values for the electron phonon interaction from thelocal density approximation we get good agreement for $\alpha$ as a function of$T_c$ and doping $\delta$ with recent experimental data in YBCO. Our resultsstrongly suggest that the large increase of $\alpha$ in the underdoped regionis (a) caused by the shift of electronic spectral density from low to highenergies associated with a competing phase (in our case a charge density wave)and the formation of a gap, and (b) compatible with the small electron phononcoupling constants obtained from the local density approximation. We propose asimilar explanation for the anomalous behavior of $\alpha$ in Sr dopedLa$_2$CuO$_4$ near the doping 1/8.

Isotope effect on the superconducting critical temperature of cuprates in the presence of charge order