{
  "id": "math/0604469",
  "version": "v1",
  "published": "2006-04-21T17:57:08.000Z",
  "updated": "2006-04-21T17:57:08.000Z",
  "title": "Positive solutions to nonlinear p-Laplace equations with Hardy potential in exterior domains",
  "authors": [
    "Vitali Liskevich",
    "Sofya Lyakhova",
    "Vitaly Moroz"
  ],
  "comment": "34 pages, 1 figure",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the existence and nonexistence of positive (super) solutions to the nonlinear $p$-Laplace equation $$-\\Delta_p u-\\frac{\\mu}{|x|^p}u^{p-1}=\\frac{C}{|x|^{\\sigma}}u^q$$ in exterior domains of ${\\R}^N$ ($N\\ge 2$). Here $p\\in(1,+\\infty)$ and $\\mu\\le C_H$, where $C_H$ is the critical Hardy constant. We provide a sharp characterization of the set of $(q,\\sigma)\\in\\R^2$ such that the equation has no positive (super) solutions. The proofs are based on the explicit construction of appropriate barriers and involve the analysis of asymptotic behavior of super-harmonic functions associated to the $p$-Laplace operator with Hardy-type potentials, comparison principles and an improved version of Hardy's inequality in exterior domains. In the context of the $p$-Laplacian we establish the existence and asymptotic behavior of the harmonic functions by means of the generalized Pr\\\"ufer-Transformation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-04-21T17:57:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "35B05",
      "35R45"
    ],
    "keywords": [
      "nonlinear p-laplace equations",
      "exterior domains",
      "hardy potential",
      "positive solutions",
      "asymptotic behavior"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1016/j.jde.2006.09.001",
      "journal": "Journal of Differential Equations",
      "year": 2007,
      "volume": 232,
      "number": 1,
      "pages": 212
    },
    "note": {
      "typesetting": "TeX",
      "pages": 34,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007JDE...232..212L"
    }
  }
}