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

TL;DR

This paper proves the existence of cylindrically symmetric ground-state solutions for a nonlinear curl-curl equation with non-constant coefficients, extending the class of problems with known solutions in nonlinear Maxwell equations.

Contribution

It introduces a new existence result for ground-state solutions in a nonlinear Maxwell problem with variable coefficients and symmetry, using advanced variational and rearrangement techniques.

Findings

Existence of symmetric ground-state solutions established.

Extension of solution classes for nonlinear Maxwell equations.

Application of advanced variational methods and inequalities.

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.