{
  "id": "1907.13027",
  "version": "v1",
  "published": "2019-07-30T15:41:47.000Z",
  "updated": "2019-07-30T15:41:47.000Z",
  "title": "Boundedness of stable solutions to nonlinear equations involving the $p$-Laplacian",
  "authors": [
    "Pietro Miraglio"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider stable solutions to the equation $ -\\Delta_p u =f(u) $ in a smooth bounded domain $\\Omega\\subset\\mathbb{R}^n $ for a $ C^1 $ nonlinearity $f$. Either in the radial case, or for some model nonlinearities $f$ in a general domain, stable solutions are known to be bounded in the optimal dimension range $n<p+4p/(p-1)$. In this article, under a new condition on $n$ and $p$, we establish an $ L^\\infty $ a priori estimate for stable solutions which holds for every $ f\\in C^1$. Our condition is optimal in the radial case for $n\\geq3$, whereas it is more restrictive in the nonradial case. This work improves the known results in the topic and gives a unified proof for the radial and the nonradial cases. The existence of an $L^\\infty$ bound for stable solutions holding for all $C^1$ nonlinearities when $n<p+4p/(p-1)$ has been an open problem over the last twenty years. A forthcoming paper by Cabr\\'e, Sanch\\'on, and the author will solve it when $p>2$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-30T15:41:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "stable solutions",
      "nonlinear equations",
      "boundedness",
      "nonradial case",
      "nonlinearity"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}