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

Abstract

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 ⊂ ℝ³, where ∇× denotes the curl operator in ℝ³. 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.