{
  "id": "1812.03418",
  "version": "v1",
  "published": "2018-12-09T02:42:14.000Z",
  "updated": "2018-12-09T02:42:14.000Z",
  "title": "Wolff's inequality for intrinsic nonlinear potentials and quasilinear elliptic equations",
  "authors": [
    "Igor E. Verbitsky"
  ],
  "comment": "25 pages",
  "categories": [
    "math.AP",
    "math.CA"
  ],
  "abstract": "We prove an analogue of Wolff's inequality for the so-called intrinsic nonlinear potentials associated with the quasilinear elliptic equation \\[ -\\Delta_{p} u = \\sigma u^{q} \\quad \\text{in} \\;\\; \\mathbb{R}^n, \\] in the sub-natural growth case $0<q< p-1$, where $\\Delta_{p}u = \\text{div}( |\\nabla u|^{p-2} \\nabla u )$ is the $p$-Laplacian, and $\\sigma$ is a nonnegative measurable function (or measure) on $\\mathbb{R}^n$. As an application, we give a necessary and sufficient condition for the existence of a positive solution $u \\in L^{r}(\\mathbb{R}^{n})$ ($0<r<\\infty$) to this problem, which was open even in the case $p=2$. Our version of Wolff's inequality for intrinsic nonlinear potentials relies on a new characterization of discrete Littlewood-Paley spaces $f^{p, q}(\\sigma)$ defined in terms of characteristic functions of dyadic cubes in $\\mathbb{R}^n$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-09T02:42:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J92",
      "42B37",
      "35J20"
    ],
    "keywords": [
      "quasilinear elliptic equation",
      "wolffs inequality",
      "intrinsic nonlinear potentials relies",
      "sub-natural growth case",
      "discrete littlewood-paley spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}