{
  "id": "1510.03879",
  "version": "v1",
  "published": "2015-10-13T20:21:19.000Z",
  "updated": "2015-10-13T20:21:19.000Z",
  "title": "Compactness results for the $p$-Laplace equation",
  "authors": [
    "Marino Badiale",
    "Michela Guida",
    "Sergio Rolando"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:1403.3803",
  "categories": [
    "math.AP",
    "math.FA"
  ],
  "abstract": "Given $1<p<N$ and two measurable functions $V(r)\\geq 0$ and $K(r)>0$, $r>0$, we define the weighted spaces \\[ W=\\left\\{ u\\in D^{1,p}(\\mathbb{R}^N):\\int_{\\mathbb{R}^N}V\\left(\\left|x\\right|\\right) \\left|u\\right|^p dx<\\infty \\right\\} , \\quad L_{K}^q =L^q(\\mathbb{R}^N,K\\left( \\left| x\\right| \\right) dx) \\] and study the compact embeddings of the radial subspace of $W$ into $L_{K}^{q_1}+L_{K}^{q_2}$, and thus into $L_{K}^q$ ($=L_{K}^q+L_{K}^q$) as a particular case. Both exponents $q_1,q_2,q$ greater and lower than $p$ are considered. Our results do not require any compatibility between how the potentials $V$ and $K$ behave at the origin and at infinity, and essentially rely on power type estimates of their relative growth, not of the potentials separately.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-13T20:21:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E35",
      "46E30",
      "35J92",
      "35J20"
    ],
    "keywords": [
      "laplace equation",
      "compactness results",
      "power type estimates",
      "compact embeddings",
      "potentials"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151003879B"
    }
  }
}