Hairy Reissner-Nordström black holes with asymmetric vacua

We minimally coupled a scalar potential V(φ) 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 φH is nontrivial at the horizon. The values of q are bounded for 0≤q≤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 φH and q, for instance, the gradient of scalar field at the horizon can increase very sharply when q→1 and φH is small, but its gradient can be very small which is independent of q when φH is large. Furthermore, the weak energy condition of hairy RNBHs, particularly at the horizon can be satisfied when $q>0$.