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Nonlinear time-harmonic Maxwell equations in a bounded domain: lack of compactness. (English) Zbl 1402.35271

Summary: We survey recent results on ground and bound state solutions \(E:\Omega\to\mathbb{R}^3\) of the problem \[ \begin{cases}\nabla\times(\nabla\times E)+\lambda E=|E|^{p-2}E&\text{in }\Omega,\\ \nu\times E=0&\text{on }\partial\Omega\end{cases} \] 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 \(\mathfrak{R}\{E(x)\mathrm{e}^{i\omega t}\}\) in a nonlinear isotropic material \(\Omega\) with \(\lambda=-\mu\varepsilon\omega^2\leqslant0\), 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 \(2<p\leqslant2^\ast=6\) comes from the nonlinear polarization and the 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.

MSC:

35Q60 PDEs in connection with optics and electromagnetic theory
35J20 Variational methods for second-order elliptic equations
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
35B33 Critical exponents in context of PDEs
78A25 Electromagnetic theory (general)
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