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

Abstract

We consider the nonlinear curl-curl problem $\nabla \times \nabla \times U+V(x)U=f(x,|U{|}^2)U$ in ${\mathbb{R}}^3$ 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 due to Lions, new rearrangement type inequalities from Brock and the recent extension of the Nehari-manifold technique by Szulkin and Weth.

13 pages