Hairy Reissner-Nordström black holes with asymmetric vacua

We minimally coupled a scalar potential $V(\ensuremath{\phi})$ with asymmetric vacua to the Einstein gravity to numerically construct the hairy Reissner-Nordstr"om black hole (RNBH) as a direct generalization of RNBHs to possess scalar hair. By fixing the electric charge to mass ratio $q$, a branch of hairy RNBHs bifurcates from the RNBH when the scalar field ${\ensuremath{\phi}}_{H}$ is nontrivial at the horizon. The values of $q$ are bounded for $0\ensuremath{\le}q\ensuremath{\le}1$, which contrast to a class of hairy black holes with $q>1$ in the Einstein-Maxwell-scalar theory. We find that the profiles of solutions affected by the competition between the strength of ${\ensuremath{\phi}}_{H}$ and $q$, for instance, the gradient of scalar field at the horizon can increase very sharply when $q\ensuremath{\rightarrow}1$ and ${\ensuremath{\phi}}_{H}$ is small, but its gradient can be very small which is independent of $q$ when ${\ensuremath{\phi}}_{H}$ is large. Furthermore, the weak energy condition of hairy RNBHs, particularly at the horizon can be satisfied when $q>0$.