{
  "id": "2408.12954",
  "version": "v1",
  "published": "2024-08-23T10:04:25.000Z",
  "updated": "2024-08-23T10:04:25.000Z",
  "title": "Existence results for a borderline case of a class of p-Laplacian problems",
  "authors": [
    "Anna Maria Candela",
    "Kanishka Perera",
    "Addolorata Salvatore"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "The aim of this paper is investigating the existence of at least one nontrivial bounded solution of the new asymptotically ``linear'' problem \\[ \\left\\{ \\begin{array}{ll} - {\\rm div} \\left[\\left(A_0(x) + A(x) |u|^{ps}\\right) |\\nabla u|^{p-2} \\nabla u\\right] + s\\ A(x) |u|^{ps-2} u\\ |\\nabla u|^p &\\\\ \\qquad\\qquad\\qquad =\\ \\mu |u|^{p (s + 1) -2} u + g(x,u) & \\hbox{in $\\Omega$,}\\\\ u = 0 & \\hbox{on $\\partial{\\Omega}$,} \\end{array}\\right.\\] where $\\Omega$ is a bounded domain in $\\mathbb{R}^N$, $N \\ge 2$, $1 < p < N$, $s > 1/p$, both the coefficients $A_0(x)$ and $A(x)$ are in $L^\\infty(\\Omega)$ and far away from 0, $\\mu \\in \\mathbb{R}$, and the ``perturbation'' term $g(x,t)$ is a Carath\\'{e}odory function on $\\Omega \\times \\mathbb{R}$ which grows as $|t|^{r-1}$ with $1\\le r < p (s + 1)$ and is such that $g(x,t) \\approx \\nu |t|^{p-2} t$ as $t \\to 0$. By introducing suitable thresholds for the parameters $\\nu$ and $\\mu$, which are related to the coefficients $A_0(x)$, respectively $A(x)$, under suitable hypotheses on $g(x,t)$, the existence of a nontrivial weak solution is proved if either $\\nu$ is large enough with $\\mu$ small enough or $\\nu$ is small enough with $\\mu$ large enough. Variational methods are used and in the first case a minimization argument applies while in the second case a suitable Mountain Pass Theorem is used.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-23T10:04:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J62",
      "35J92",
      "47J30",
      "35Q55",
      "58E30"
    ],
    "keywords": [
      "borderline case",
      "existence results",
      "p-laplacian problems",
      "nontrivial weak solution",
      "minimization argument applies"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}