{
  "id": "2101.02368",
  "version": "v1",
  "published": "2021-01-07T05:05:01.000Z",
  "updated": "2021-01-07T05:05:01.000Z",
  "title": "Bilateral estimates of solutions to quasilinear elliptic equations with sub-natural growth terms",
  "authors": [
    "Igor Verbitsky"
  ],
  "comment": "18 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study quasilinear elliptic equations of the type $-\\Delta_{p} u = \\sigma u^{q} + \\mu \\; \\; \\text{in} \\;\\; \\bf{R}^n$ in the case $0<q< p-1$, where $\\mu$ and $\\sigma$ are nonnegative measurable functions, or locally finite measures, and $\\Delta_{p}u= \\text{div}(|\\nabla u|^{p-2}\\nabla u)$ is the $p$-Laplacian. Similar equations with more general local and nonlocal operators in place of $\\Delta_{p}$ are treated as well. We obtain existence criteria and global bilateral pointwise estimates for all positive solutions $u$: $$ u(x) \\approx (\\bf{W}_p \\sigma(x))^{\\frac{p-q}{p-q-1}} + \\bf{K}_{p,q} \\sigma(x) + \\bf{W}_p \\mu (x), \\quad x \\in \\bf{R}^n,$$ where $\\bf{W}_p$ and $\\bf{K}_{p, q}$ are, respectively, the Wolff potential and the intrinsic Wolff potential, with the constants of equivalence depending only on $p$, $q$ and $n$. The contributions of $\\mu$ and $\\sigma$ in these pointwise estimates are totally separated, which is a new phenomenon even when $p=2$. In the homogeneous case $\\mu=0$, such estimates were obtained earlier by a different method only for minimal positive solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-07T05:05:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J92",
      "42B37",
      "35J20"
    ],
    "keywords": [
      "sub-natural growth terms",
      "bilateral estimates",
      "study quasilinear elliptic equations",
      "intrinsic wolff potential",
      "global bilateral pointwise estimates"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}