Nonlinear time-harmonic Maxwell equations in a bounded domain: Lack of compactness

We survey recent results on ground and bound state solutions (E:Omegarightarrowmathbb{R}^3) of the problem

$${ begin{array}{*{20}{c}}{nabla times left( {nabla times E} right) + lambda E = {{left| E right|}^{p - 2}}EinOmega ,} {v times E = 0onpartial Omega }end{array}$$

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}^{iomegait{t}}}) in a nonlinear isotropic material Ω with (lambda=-muvarepsilonomega^2leqslant0), 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 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.