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Science China Mathematics

, Volume 61, Issue 11, pp 1963–1970 | Cite as

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

  • Jarosław MederskiEmail author
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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.

Keywords

time-harmonic Maxwell equations perfect conductor ground state variational methods strongly indefinite functional Nehari-Pankov manifold Brezis-Nirenberg problem critical exponent 

MSC(2010)

35Q60 35J20 58E05 35B33 78A25 

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Notes

Acknowledgements

This work was supported by the National Science Centre of Poland (Grant No. 2013/09/B/ST1/01963).

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Copyright information

© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of MathematicsPolish Academy of SciencesWarszawaPoland
  2. 2.Faculty of Mathematics and Computer ScienceNicolaus Copernicus UniversityToruńPoland

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