Higher dimensional Reissner-Nordström black holes supporting static scalar shells

We analytically study scalarization of higher-dimensional charged Reissner-Nordström (RN) black hole. It is shown that static massive scalar field which is non-minimally coupled to Gauss-Bonnet invariant can be supported by higher-dimensional black hole in super-critical charge regime $Q\mathrm{/}M\ge {\bar{C}}_d$ with $Q,M$ charge and mass of the black hole and ${\bar{C}}_d$ some unitless spacetime dimension-dependent quantity. Moreover, we show that the static massive scalar shell can be quite thin in the large mass regime $\mu M^{\frac{1}{d-3}}\gg 1$ with $\mu$ mass of the scalar field.