We present high resolution 240 and 607 MHz GMRT radio observations, complemented with 74 MHz archival VLA radio observations of the Ophiuchus cluster of galaxies, whose radio mini-halo has been recently detected at 1400 MHz. We also present archival Chandra and XMM-Newton data of the Ophiuchus cluster. Our observations do not show significant radio emission from the mini-halo, hence we present upper limits to the integrated, diffuse non-thermal radio emission of the core of the Ophiuchus cluster. The XMM-Newton observations can be well explained by a two-temperature thermal model with temperatures of ~=1.8 keV and ~=9.0 keV, respectively, which confirms previous results that suggest that the innermost central region of the Ophiuchus cluster is a cooling core. We also used the XMM-Newton data to set up an upper limit to the (non-thermal) X-ray emission from the cluster. The combination of available radio and X-ray data has strong implications for the currently proposed models of the spectral energy distribution (SED) from the Ophiuchus cluster. In particular, a synchrotron+IC model is in agreement with the currently available data, if the average magnetic field is in the range (0.02-0.3) microG. A pure WIMP annihilation scenario can in principle reproduce both radio and X-ray emission, but at the expense of postulating very large boost factors from dark matter substructures, jointly with extremely low values of the average magnetic field. Finally, a scenario where synchrotron and inverse Compton emission arise from PeV electron-positron pairs (via interactions with the CMB), can be ruled out, as it predicts a non-thermal soft X-ray emission that largely exceeds the thermal Bremsstrahlung measured by INTEGRAL. ; Comment: Accepted for publication in MNRAS; 13 pages, 8 figures. Includes minor changes. Abridged abstract