Properties of Scalar Hairy Black Holes and Scalarons with Asymmetric Potential

In this paper we study the properties of black holes and scalarons in Einstein gravity when it is minimally coupled to a scalar field $\phi$ with an asymmetric potential $V(\phi )$, constructed in [Phys. Rev. D $\mathbf{\text{73}}$ (2006), 084002] a few decades ago. $V(\phi )$ has been applied in the cosmology to describe the quantum tunneling process from the false vacuum to the true vacuum and contains a local maximum, a local minimum (false vacuum) and a global minimum (true vacuum). In particular we focus on the asymptotically flat solutions, which can be constructed by fixing appropriately the local minimum of $V$. A branch of hairy black holes solutions emerge from the Schwarzschild black hole, and we study the domain of existence of such configurations. They can reach to a particle-like solution in the small horizon limit, i.e. the scalarons. We study the stability of black holes and scalarons, showing that all of them are unstable under radial perturbations.