  An important question concerning the classical solutions of the equations of
motion arising in quantum field theories at the BPS critical coupling is
whether all finite-energy solutions are necessarily BPS. In this paper we
present a study of this basic question in the context of the domain wall
equations whose potential is induced from a superpotential so that the ground
states are the critical points of the superpotential. We prove that the
definiteness of the Hessian of the superpotential suffices to ensure that all
finite-energy domain-wall solutions are BPS. We give several examples to show
that such a BPS property may fail such that non-BPS solutions exist when the
Hessian of the superpotential is indefinite.
