We survey recent results on ground and bound state solutions
\(E:\Omega\rightarrow\mathbb{R}^3\) of the problem
$$\{ \begin{array}{*{20}{c}}{\nabla \times \left( {\nabla \times E} \right) + \lambda E = {{\left| E \right|}^{p - 2}}Ein\Omega ,} \\{v \times E = 0on\partial \Omega }\end{array}$$
on a bounded Lipschitz domain Ω ? ?
3, where ?× denotes the curl operator in ?
3. The equation describes the propagation of the time-harmonic electric field
\(\mathfrak{R}\{E(x)\rm{e}^{i\omega\it{t}}\}\) in a nonlinear isotropic material Ω with
\(\lambda=-\mu\varepsilon\omega^2\leqslant0\), where
μ and
ε stand for the permeability and the linear part of the permittivity of the material. The nonlinear term
\(|E|^{p-2}E\) with
\(2<p\leqslant2^*=6\) comes from the nonlinear polarization and the boundary conditions are those for Ω surrounded by a perfect conductor. The problem has a variational structure; however the energy functional associated with the problem is strongly indefinite and does not satisfy the Palais-Smale condition. We show the underlying difficulties of the problem and enlist some open questions.