The Brezis-Nirenberg problem for the curl-curl operator

TL;DR

This paper investigates solutions to a nonlinear curl-curl equation modeling electric fields in anisotropic media, establishing existence of symmetric ground and bound states using a novel critical point theory.

Contribution

It introduces a new critical point framework to solve the nonlinear curl-curl problem, including for anisotropic media and critical exponents.

Findings

Existence of cylindrically symmetric ground states.

Finite number of symmetric bound states depending on parameters.

Applicable to anisotropic and more general media.

Abstract

We look for solutions $E:\Omega\to\mathbb{R}^3$ of the problem $$ \left\{ \begin{aligned} &\nabla\times(\nabla\times E) +\lambda E = |E|^{p-2}E &&\quad \text{in }\Omega &\nu\times E = 0 &&\quad \text{on }\partial\Omega \end{aligned} \right. $$ on a bounded Lipschitz domain $\Omega\subset\mathbb{R}^3$, where $\nabla\times$ denotes the curl operator in $\mathbb{R}^3$. The equation describes the propagation of the time-harmonic electric field $\Re\{E(x)e^{i\omega t}\}$ in a nonlinear isotropic material $\Omega$ with $\lambda=-\mu \varepsilon \omega^2\leq 0$, where $\mu$ and $\varepsilon$ stand for the permeability and the linear part of the permittivity of the material. The nonlinear term $|E|^{p-2}E$ with $p>2$ is responsible for the nonlinear polarisation of $\Omega$ and the boundary conditions are those for $\Omega$ surrounded by a perfect conductor. The problem has a variational structure and we deal with the critical values $p$, for instance, in convex domains $\Omega$ or in domains with $\mathcal{C}^{1,1}$ boundary $p=6=2^*$ is the Sobolev critical exponent and we get the quintic nonlinearity in the equation. We show that there exist a cylindrically symmetric ground state solution and a finite number of cylindrically symmetric bound states depending on $\lambda\leq 0$. We develop a new critical point theory which allows to solve the problem, and which enables us to treat more general anisotropic media as well as other variational problems.