{
  "id": "1811.10163",
  "version": "v1",
  "published": "2018-11-26T03:33:36.000Z",
  "updated": "2018-11-26T03:33:36.000Z",
  "title": "Solutions in Lebesgue spaces to nonlinear elliptic equations with sub-natural growth terms",
  "authors": [
    "Adisak Seesanea",
    "Igor E. Verbitsky"
  ],
  "comment": "20 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the existence problem for positive solutions $u \\in L^{r}(\\mathbb{R}^{n})$, $0<r<\\infty$, to 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 with $1<p<\\infty$, and $\\sigma$ is a nonnegative measurable function (or measure) on $\\mathbb{R}^n$. Our techniques rely on a study of general integral equations involving nonlinear potentials and related weighted norm inequalities. They are applicable to more general quasilinear elliptic operators such as the $\\mathcal{A}$-Laplacian $\\text{div} \\mathcal{A}(x,\\nabla u)$, and the fractional Laplacian $(-\\Delta)^{\\alpha}$ on $\\mathbb{R}^n$, as well as linear uniformly elliptic operators with bounded measurable coefficients $\\text{div}(\\mathcal{A} \\nabla u)$ on an arbitrary domain $\\Omega \\subseteq \\mathbb{R}^n$ with a positive Green function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-26T03:33:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J92",
      "35J20",
      "42B37"
    ],
    "keywords": [
      "sub-natural growth terms",
      "nonlinear elliptic equations",
      "lebesgue spaces",
      "general quasilinear elliptic operators",
      "general integral equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}