Existence of cylindrically symmetric ground states to a nonlinear curl-curl equation with non-constant coefficients

We consider the nonlinear curl-curl problem ∇ × ∇ × U + V(x)U = f (x, |U|2)U in R3 related to the nonlinear Maxwell equations with Kerr-type nonlinear material laws. We prove the existence of a symmetric ground-state type solution for a bounded, cylindrically symmetric coefficient V and subcritical cylindrically symmetric nonlinearity f . The new existence result extends the class of problems for which ground-state type solutions are known. It is based on compactness properties of symmetric functions [11, 12], new rearrangement type inequalities from [6] and the recent extension of the Nehari-manifold technique from [18].