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


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.


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


35Q60 35J20 58E05 35B33 78A25 


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


  1. 1.
    Amrouche C, Bernardi C, Dauge M, et al. Vector potentials in three-dimensional non-smooth domains. Math Methods Appl Sci, 1998, 21: 823–864MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bartsch T, Ding Y. Deformation theorems on non-metrizable vector spaces and applications to critical point theory. Math Nachr, 2006, 279: 1267–1288MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bartsch T, Mederski J. Ground and bound state solutions of semilinear time-harmonic Maxwell equations in a bounded domain. Arch Ration Mech Anal, 2015, 215: 283–306MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bartsch T, Mederski J. Nonlinear time-harmonic Maxwell equations in an anisotropic bounded medium. J Funct Anal, 2017, 272: 4304–4333MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Bartsch T, Mederski J. Nonlinear time-harmonic Maxwell equations in domains. J Fixed Point Theory Appl, 2017, 19: 959–986MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Benci V, Rabinowitz P H. Critical point theorems for indefinite functionals. Invent Math, 1979, 52: 241–273MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Birman M S, Solomyak M Z. L2-theory of the Maxwell operator in arbitrary bounded domains. Russian Math Surveys, 1987, 42: 75–96CrossRefzbMATHGoogle Scholar
  8. 8.
    Brezis H, Nirenberg L. Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents. Comm Pure Appl Math, 1983, 36: 437–477MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Costabel M. A remark on the regularity of solutions of Maxwell’s equations on Lipschitz domains. Math Methods Appl Sci, 1990, 12: 365–368MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Dörfler W, Lechleiter A, Plum M, et al. Photonic Crystals: Mathematical Analysis and Numerical Approximation. Basel: Springer, 2012zbMATHGoogle Scholar
  11. 11.
    Kirsch A, Hettlich F. The Mathematical Theory of Time-Harmonic Maxwell’s Equations: Expansion-, Integral-, and Variational Methods. New York: Springer, 2015zbMATHGoogle Scholar
  12. 12.
    Kryszewski W, Szulkin A. Generalized linking theorem with an application to semilinear Schrödinger equation. Adv Differential Equations, 1998, 3: 441–472MathSciNetzbMATHGoogle Scholar
  13. 13.
    Leis R. Zur Theorie elektromagnetischer Schwingungen in anisotropen inhomogenen Medien. Math Z, 1968, 106: 213–224MathSciNetCrossRefGoogle Scholar
  14. 14.
    Mederski J. Ground states of time-harmonic semilinear Maxwell equations in ℝ3 with vanishing permittivity. Arch Ration Mech Anal, 2015, 218: 825–861MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Mederski J. Solutions to a nonlinear Schrödinger equation with periodic potential and zero on the boundary of the spectrum. Topol Methods Nonlinear Anal, 2015, 46: 755–771MathSciNetzbMATHGoogle Scholar
  16. 16.
    Mederski J. Ground states of a system of nonlinear Schrödinger equations with periodic potentials. Comm Partial Differential Equations, 2016, 41: 1426–1440MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Mederski J. The Brezis-Nirenberg problem for the curl-curl operator. J Funct Anal, 2018, 274: 1345–1380MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Mitrea M. Sharp Hodge decompositions, Maxwell’s equations, and vector Poisson problems on nonsmooth, threedimensional Riemannian manifolds. Duke Math J, 2004, 125: 467–547MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Monk P. Finite Element Methods for Maxwell’s Equations. Oxford: Oxford University Press, 2003CrossRefzbMATHGoogle Scholar
  20. 20.
    Pankov A. Periodic Nonlinear Schrödinger Equation with Application to Photonic Crystals. Milan J Math, 2005, 73: 259–287MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Picard R, Weck N, Witsch K-J. Time-harmonic Maxwell equations in the exterior of perfectly conducting, irregular obstacles. Analysis (Munich), 2001, 21: 231–263MathSciNetzbMATHGoogle Scholar
  22. 22.
    Rabinowitz P. Minimax Methods in Critical Point Theory with Applications to Differential Equations. CBMS Regional Conference Series in Mathematics, vol. 65. Providence: Amer Math Soc, 1986Google Scholar
  23. 23.
    Saleh B E A, Teich M C. Fundamentals of Photonics, 2nd ed. New York: Wiley, 2007Google Scholar
  24. 24.
    Struwe M. Variational Methods. New York: Springer-Verlag, 2008zbMATHGoogle Scholar
  25. 25.
    Stuart C A. Self-trapping of an electromagnetic field and bifurcation from the essential spectrum. Arch Ration Mech Anal, 1991, 113: 65–96MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Szulkin A, Weth T. Ground state solutions for some indefinite variational problems. J Funct Anal, 2009, 257: 3802–3822MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Zeng X. Cylindrically symmetric ground state solutions for curl-curl equations with critical exponent. Z Angew Math Phys, 2017, 68: 135MathSciNetCrossRefzbMATHGoogle Scholar

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© 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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