{
  "id": "1109.5142",
  "version": "v2",
  "published": "2011-09-23T17:47:48.000Z",
  "updated": "2013-05-24T03:06:42.000Z",
  "title": "Effect of weights on stable solutions of a quasilinear elliptic equation",
  "authors": [
    "Mostafa Fazly"
  ],
  "comment": "The original version posted on arxiv in 2011 and was not submitted for publication. This note is accepted in Nonlinear Dynamics and Systems Theory (Invited paper for a special issue)",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this note, we study Liouville theorems for the stable and finite Morse index weak solutions of the quasilinear elliptic equation $-\\Delta_p u= f(x) F(u) $ in $\\mathbb{R}^n$ where $p\\ge 2$, $0\\le f\\in C(\\mathbb{R}^n)$ and $F\\in C^1(\\mathbb{R})$. We refer to $f(x)$ as {\\it weight} and to $F(u)$ as {\\it nonlinearity}. The remarkable fact is that if the weight function is bounded from below by a strict positive constant that is $0<C\\le f$ then it does not have much impact on the stable solutions, however, a nonnegative weight that is $0\\le f$ will push certain critical dimensions. This analytical observation has potential to be applied in various models to push certain well-known critical dimensions. For a general nonlinearity $F\\in C^1(\\mathbb{R})$ and $f(x)=|x|^\\alpha$, we prove Liouville theorems in dimensions $n\\le \\frac{4(p+\\alpha)}{p-1}+p$, for bounded radial stable solutions. For specific nonlinearities $F(u)=e^u$, $u^q$ where $q>p-1$ and $-u^{q}$ where $q<0$, known as the Gelfand, the Lane-Emden and the negative exponent nonlinearities, respectively, we prove Liouville theorems for both radial finite Morse index (not necessarily bounded) and stable (not necessarily radial nor bounded) solutions.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-05-24T03:06:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "quasilinear elliptic equation",
      "stable solutions",
      "liouville theorems",
      "finite morse index weak solutions",
      "radial finite morse index"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1109.5142F"
    }
  }
}