Ground States of a Nonlinear Curl-Curl Problem in Cylindrically Symmetric Media

TL;DR

This paper investigates the existence of ground state solutions to a nonlinear Maxwell-related curl-curl problem in cylindrically symmetric media, using variational methods and concentration compactness principles.

Contribution

It establishes conditions for the existence of ground states in both defocusing and focusing cases under cylindrical symmetry assumptions.

Findings

Ground states exist for strongly defocusing media with large negative $\Gamma$ at infinity.

Ground states are obtained for focusing media when zero is outside the spectrum of the linear operator.

Examples of cylindrically symmetric functions $V$ satisfying the spectral conditions are provided.

Abstract

We consider the nonlinear curl-curl problem $\nabla\times\nabla\times U + V(x) U= \Gamma(x)|U|^{p-1}U$ in $\mathbb{R}^3$ related to the nonlinear Maxwell equations for 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 $H(\mathrm{curl};\mathbb{R}^3)$. Under a cylindrical symmetry assumption on the functions $V$ and $\Gamma$ the variational problem can be posed in a symmetric subspace of $H(\mathrm{curl};\mathbb{R}^3)$. For a strongly defocusing case $\mathrm{esssup}\, \Gamma <0$ with large negative values of $\Gamma$ at infinity we obtain ground states by the direct minimization method. For the focusing case $\mathrm{essinf}\, \Gamma >0$ the concentration compactness principle produces ground states under the assumption that zero lies outside the spectrum of the linear operator $\nabla \times \nabla \times +V(x)$. Examples of cylindrically symmetric functions $V$ are provided for which this holds.