Ground states of a nonlinear curl-curl problem in cylindrically symmetric media

We consider the nonlinear curl-curl problem del x del x U + V (x)U = Gamma(x)vertical bar U vertical bar(p-1) U in R-3 related to the Kerr nonlinear Maxwell equations for fully localized monochromatic fields. We search for solutions as minimizers (ground states) of the corresponding energy functional defined on subspaces (defocusing case) or natural constraints (focusing case) of H(curl; R-3). Under a cylindrical symmetry assumption corresponding to a photonic fiber geometry on the functions V and G the variational problem can be posed in a symmetric subspace of H(curl; R-3). For a defocusing case sup Gamma < 0 with large negative values of G at infinity we obtain ground states by the direct minimization method. For the focusing case inf Gamma > 0 the concentration compactness principle produces ground states under the assumption that zero lies outside the spectrum of the linear operator del x del x + V(x). Examples of cylindrically symmetric functions V are provided for which this holds.