{
  "id": "1603.09265",
  "version": "v1",
  "published": "2016-03-30T16:20:44.000Z",
  "updated": "2016-03-30T16:20:44.000Z",
  "title": "Moderate solutions of semilinear elliptic equations with Hardy potential under minimal restrictions on the potential",
  "authors": [
    "Moshe Marcus",
    "Vitaly Moroz"
  ],
  "comment": "22 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study semilinear elliptic equations with Hardy potential $\\mathrm{(E)} \\; -L_\\mu u+u^q=0$ in a bounded smooth domain $\\Omega\\subset \\mathbb R^N$. Here $q>1$, $L_\\mu=\\Delta+\\frac{\\mu}{\\delta_\\Omega^2}$ and $\\delta_\\Omega(x)=\\mathrm{dist}(x,\\partial\\Omega)$. Assuming that $0\\leq \\mu<C_H(\\Omega)$, boundary value problems with measure data and discrete boundary singularities for positive solutions of $\\mathrm{(E)}$ have been studied earlier. In the present paper we study these problems, in arbitrary domains, assuming only $-\\infty<\\mu<1/4$, even if $C_H(\\Omega)<1/4$. We recall that $C_H(\\Omega)\\leq 1/4$ and, in general, strict inequality holds. The key to our study is the fact that, if $\\mu<1/4$ then in smooth domains there exist local $L_\\mu$-superharmonic functions in a neighborhood of $\\partial\\Omega$ (even if $C_H(\\Omega)<1/4$). Using this fact we extend the notion of normalized boundary trace to arbitrary domains, provided that $\\mu<1/4$. Further we study the b.v.p. with normalized boundary trace $\\nu$ in the space of positive finite measures on $\\partial\\Omega$. We show that existence depends on two critical values of the exponent $q$ and discuss the question of uniqueness. Part of the paper is devoted to the study of the linear operator: properties of local $L_\\mu$ subharmonic and superharmonic functions and the related notion of moderate solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-03-30T16:20:44.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "35J75",
      "35A20",
      "31B35"
    ],
    "keywords": [
      "hardy potential",
      "moderate solutions",
      "minimal restrictions",
      "normalized boundary trace",
      "superharmonic functions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160309265M"
    }
  }
}