Nonlinear time-harmonic Maxwell equations in domains

Abstract

The search for time-harmonic solutions of nonlinear Maxwell equations in the absence of charges and currents leads to the elliptic equation

$$\begin{aligned} \nabla \times \left( \mu (x)^{-1}\nabla \times u\right) - \omega ^2\varepsilon (x)u = f(x,u) \end{aligned}$$

for the field \(u:\Omega \rightarrow \mathbb {R}^3\) in a domain \(\Omega \subset \mathbb {R}^3\). Here, \(\varepsilon (x) \in \mathbb {R}^{3\times 3}\) is the (linear) permittivity tensor of the material, and \(\mu (x) \in \mathbb {R}^{3\times 3}\) denotes the magnetic permeability tensor. The nonlinearity \(f:\Omega \times \mathbb {R}^3\rightarrow \mathbb {R}^3\) comes from the nonlinear polarization. If \(f=\nabla _uF\) is a gradient, then this equation has a variational structure. The goal of this paper is to give an introduction to the problem and the variational approach, and to survey recent results on ground and bound state solutions. It also contains refinements of known results and some new results.

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Correspondence to Thomas Bartsch.

Additional information

Dedicated to Paul Rabinowitz.

J. Mederski was partially supported by the National Science Centre, Poland (Grant No. 2014/15/D/ST1/03638).

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Bartsch, T., Mederski, J. Nonlinear time-harmonic Maxwell equations in domains. J. Fixed Point Theory Appl. 19, 959–986 (2017). https://doi.org/10.1007/s11784-017-0409-1

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Keywords

  • Time-harmonic Maxwell equations
  • perfect conductor
  • anisotropic media
  • uniaxial media
  • nonlinear material
  • ground state
  • variational methods
  • strongly indefinite functional
  • Nehari–Pankov manifold

Mathematics Subject Classification

  • Primary: 35Q60
  • Secondary: 35J20
  • 58E05
  • 78A25