{
  "id": "1708.02413",
  "version": "v1",
  "published": "2017-08-08T09:09:19.000Z",
  "updated": "2017-08-08T09:09:19.000Z",
  "title": "Compactness properties and ground states for the affine Laplacian",
  "authors": [
    "Ian Schindler",
    "Cyril Tintarev"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "The paper studies compactness properties of the affine Sobolev inequality of Gaoyong Zhang et al in the case $p=2$, and existence and regularity of related minimizers, in particular, solutions to the nonlocal Dirichlet problems \\[ -\\sum_{i,j=1}^{N}(A^{-1}[u])_{ij}\\frac{\\partial^2u}{\\partial x_i\\partial x_j}=f \\mbox{ in }\\Omega\\subset\\mathbb R^N, \\] and \\[ -\\sum_{i,j=1}^{N}(A^{-1}[u])_{ij}\\frac{\\partial^2u}{\\partial x_i\\partial x_j}=u^{q-1}\\,,\\quad u>0,\\mbox{ in }\\Omega\\subset\\mathbb R^N, \\] where $A_{ij}[u]=\\int_\\Omega\\frac{\\partial u}{\\partial x_i}\\frac{\\partial u}{\\partial x_j}\\mathrm{d}x$ and $q\\in(2,\\frac{2N}{N-2})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-08-08T09:09:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E30",
      "46E35",
      "35J20",
      "35J99",
      "35H99"
    ],
    "keywords": [
      "ground states",
      "affine laplacian",
      "paper studies compactness properties",
      "affine sobolev inequality",
      "nonlocal dirichlet problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}