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

Abstract

We survey recent results on ground and bound state solutions $E:\Omega\to\mathbb{R}^3$ of the problemon 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 $\Re\{E(x){\rm~e}^{{\rm~i}\omega~t}\}$ in a nonlinear isotropic material $\Omega$ with $\lambda=-\mu~\varepsilon~\omega^2\leq~0$, where $\mu$ and $\varepsilon$ stand forthe permeability and the linear part of the permittivity of the material. The nonlinear term $|E|^{p-2}E$ with $2<p\leq~2^*=6$ comes from the nonlinear polarization andthe boundary conditions are those for $\Omega$ 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.