  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.
