Stability analysis of a charged black hole with a nonlinear complex scalar field

It has been shown recently that the charged black hole can be scalarized if Maxwell field minimally couples with a complex scalar which nonnegative nonlinear potential. We firstly prove such scalarization cannot result of continuous phase transition for general Furthermore, we numerically find it is possible RN will by first order spontaneously and extreme not stable in micro-canonical ensemble. In addition, considering massless perturbation, compute quasi-normal modes results imply only favored thermodynamics but also kinetically against perturbation at linear level. Our numerical definitely gives negative answer to Penrose-Gibbons conjecture two new versions Penrose inequality case are suggested.