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In this paper we study several aspects of extremal spherical symmetric blackhole solutions of four dimensional N=1 supergravity coupled to vector andchiral multiplets with the scalar potential turned on. In the asymptotic regionthe complex scalars are fixed and regular which can be viewed as the criticalpoints of the black hole and the scalar potentials with vanishing scalarcharges. It follows that the asymptotic geometries are of a constant andnon-zero scalar curvature which are generally not Einstein. These spaces couldalso correspond to the near horizon geometries which are the product spaces ofa two anti-de Sitter surface and the two sphere if the value of the scalars inboth regions coincides. In addition, we prove the local existence ofnon-trivial radius dependent complex scalar fields which interpolate betweenthe horizon and the asymptotic region. We finally give some simple${\lC}^{n}$-models with both linear superpotential and gauge couplings.

Some Aspects of Spherical Symmetric Extremal Dyonic Black Holes in 4d N=1 Supergravity