Existence of Static Dyonic Black Holes in $4d$ $N=1$ Supergravity With Finite Energy

We prove the existence and the uniqueness of the static dyonic black holes in four dimensional $N=1$ supergravity theory coupled vector and scalar multiplets. We set the near-horizon geometry to be a product of two Einstein surfaces, whereas the asymptotic geometry has to be a space of constant scalar curvature. Using these data, we show that there exist a unique solution for scalar fields which interpolates these regions.