{
  "id": "1911.03908",
  "version": "v1",
  "published": "2019-11-10T11:37:48.000Z",
  "updated": "2019-11-10T11:37:48.000Z",
  "title": "Existence of radial bounded solutions for some quasilinear elliptic equations in R^N",
  "authors": [
    "Anna Maria Candela",
    "Addolorata Salvatore"
  ],
  "doi": "10.1016/j.na.2019.111625",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the quasilinear equation \\[(P)\\qquad - {\\rm div} (A(x,u) |\\nabla u|^{p-2} \\nabla u) + \\frac1p\\ A_t(x,u) |\\nabla u|^p + |u|^{p-2}u\\ =\\ g(x,u) \\qquad \\hbox{in ${\\mathbb R}^N$,} \\] with $N\\ge 3$, $p > 1$, where $A(x,t)$, $A_t(x,t) = \\frac{\\partial A}{\\partial t}(x,t)$ and $g(x,t)$ are Carath\\'eodory functions on ${\\mathbb R}^N \\times {\\mathbb R}$. Suitable assumptions on $A(x,t)$ and $g(x,t)$ set off the variational structure of $(P)$ and its related functional ${\\cal J}$ is $C^1$ on the Banach space $X = W^{1,p}({\\mathbb R}^N) \\cap L^\\infty({\\mathbb R}^N)$. In order to overcome the lack of compactness, we assume that the problem has radial symmetry, then we look for critical points of ${\\cal J}$ restricted to $X_r$, subspace of the radial functions in $X$. Following an approach which exploits the interaction between $\\|\\cdot\\|_X$ and the norm on $W^{1,p}({\\mathbb R}^N)$, we prove the existence of at least one weak bounded radial solution of $(P)$ by applying a generalized version of the Ambrosetti-Rabinowitz Mountain Pass Theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-10T11:37:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J20",
      "35J92",
      "35Q55",
      "58E05"
    ],
    "keywords": [
      "quasilinear elliptic equations",
      "radial bounded solutions",
      "ambrosetti-rabinowitz mountain pass theorem",
      "weak bounded radial solution",
      "radial functions"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "Elsevier"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}