{
  "id": "1609.03989",
  "version": "v1",
  "published": "2016-09-13T19:30:48.000Z",
  "updated": "2016-09-13T19:30:48.000Z",
  "title": "The Brezis-Nirenberg problem for the curl-curl operator",
  "authors": [
    "Jarosław Mederski"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We look for solutions $E:\\Omega\\to\\mathbb{R}^3$ of the problem \\[ \\left\\{ &\\nabla\\times(\\nabla\\times E) +\\lambda E = |E|^{p-2}E &&\\quad \\text{in }\\Omega &\\nu\\times E = 0 &&\\quad \\text{on }\\partial\\Omega \\right. \\] 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 $\\Re\\{E(x)e^{i\\omega t}\\}$ in a nonlinear isotropic material $\\Omega$ with $\\lambda=-\\mu \\varepsilon \\omega^2\\leq 0$, 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 $p>2$ is responsible for the nonlinear polarisation of $\\Omega$ and the boundary conditions are those for $\\Omega$ surrounded by a perfect conductor. The problem has a variational structure and we deal with the critical values $p$, for instance, in convex domains $\\Omega$ or in domains with $\\mathcal{C}^{1,1}$ boundary $p=6=2^*$ is the Sobolev critical exponent and we get the quintic nonlinearity in the equation. We show that there exist a cylindrically symmetric ground state solution and at least finite number of cylindrically symmetric bound states depending on $\\lambda\\leq 0$. We develop a new critical point theory which allows to solve the problem, and which enables us to treat more general anisotropic media as well as other variational problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-09-13T19:30:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "brezis-nirenberg problem",
      "curl-curl operator",
      "cylindrically symmetric ground state solution",
      "general anisotropic media",
      "time-harmonic electric field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}