{
  "id": "1403.7572",
  "version": "v1",
  "published": "2014-03-29T00:23:20.000Z",
  "updated": "2014-03-29T00:23:20.000Z",
  "title": "A Meshkov-type construction for the borderline case",
  "authors": [
    "Blair Davey"
  ],
  "comment": "15 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We construct functions $u: \\mathbb{R}^2 \\to \\mathbb{C}$ that satisfy an elliptic eigenvalue equation of the form $-\\Delta u + W \\cdot \\nabla u + V u = \\lambda u$, where $\\lambda \\in \\mathbb{C}$, and $V$ and $W$ satisfy $|V(x)| \\lesssim <x>^{-N}$, and $|W(x)| \\lesssim <x>^{-P}$, with $\\min\\{N, P\\} = 1/2$. For $|x|$ sufficiently large, these solutions satisfy $|u(x)| \\lesssim \\exp(- c|x|)$. In the author's previous work, examples of solutions over $\\mathbb{R}^2$ were constructed for all $N, P$ such that $\\min\\{N,P\\} \\in [0, 1/2)$. These solutions were shown to have the optimal rate of decay at infinity. A recent result of Lin and Wang shows that the constructions presented in this note for the borderline case of $\\min\\{N, P\\} = 1/2$ also have the optimal rate of decay at infinity.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-29T00:23:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J10",
      "35J15",
      "35B60"
    ],
    "keywords": [
      "borderline case",
      "meshkov-type construction",
      "optimal rate",
      "elliptic eigenvalue equation",
      "construct functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1403.7572D"
    }
  }
}