Stability of higher dimensional Reissner-Nordström-anti-de Sitter black holes

We investigate stability of the $D$-dimensional Reissner-Nordström-anti-de Sitter metrics as solutions of the Einstein-Maxwell equations. We have shown that asymptotically anti-de Sitter (AdS) black holes are dynamically stable for all values of charge and anti-de Sitter radius in $D=5,6\dots 11$ dimensional space-times. This does not contradict dynamical instability of RNAdS black holes found by Gubser in $\mathcal{N}=8$ gauged supergravity, because the latter instability comes from the tachyon mode of the scalar field, coupled to the system. Asymptotically AdS black holes are known to be thermodynamically unstable for some region of parameters, yet, as we have shown here, they are stable against gravitational perturbations.