  It is show that one can derive a novel BPS bound for the gauged
Non-Linear-Sigma-Model (NLSM) Maxwell theory in (3+1) dimensions which can
actually be saturated. Such novel bound is constructed using Hamilton-Jacobi
equation from classical mechanics. The configurations saturating the bound
represent Hadronic layers possessing both Baryonic charge and magnetic flux.
However, unlike what happens in the more common situations, the topological
charge which appears naturally in the BPS bound is a non-linear function of the
Baryonic charge. This BPS bound can be saturated when the surface area of the
layer is quantized. The far-reaching implications of these results are
discussed. In particular, we determine the exact relation between the magnetic
flux and the Baryonic charge as well as the critical value of the Baryonic
chemical potential beyond which these configurations become thermodynamically
unstable.
