{
  "id": "2201.12651",
  "version": "v1",
  "published": "2022-01-29T20:26:21.000Z",
  "updated": "2022-01-29T20:26:21.000Z",
  "title": "Existence results for singular elliptic problem involving a fractional p-Laplacian",
  "authors": [
    "Hanaa Achour",
    "Sabri Bensid"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this article, the problems to be studied are the following \\leqnomode \\begin{equation*} \\label{p} \\left\\{\\begin{array}{ll} (-\\Delta )_p^s u \\pm \\dfrac{|u|^{p-2}u}{|x|^{sp}} = \\lambda f(x,u) & \\quad \\mbox{in }\\ \\Omega\\\\[0.3cm] u= 0 & \\quad \\mbox{on }\\ \\mathbb{R}^N \\setminus \\Omega,\\tag{P$_{\\pm}$} \\end{array} \\right. \\end{equation*} \\reqnomode where $\\Omega$ is a bounded regular domain in $\\mathbb{R}^N(N\\geq 2)$ containing the origin, $p>1$, $s\\in(0,1)$, $(N>ps)$, $\\lambda>0$, $f : \\Omega \\times \\mathbb{R} \\longrightarrow \\mathbb{R}$ is a Carath\\'eodory function satisfying a suitable growth condition and $(-\\Delta )_p^s$ is the fractional p-Laplacian defined as $$(-\\Delta )_{p}^{s} u(x) = \\displaystyle 2 \\lim_{\\varepsilon \\rightarrow 0} \\int_{\\mathbb{R}^N \\setminus B_{\\varepsilon}(x)} \\dfrac{\\vert u(x)-u(y) \\vert^{p-2}(u(x)-u(y))}{\\vert x-y \\vert^{N+sp}} ~dy, ~~~~ x \\in \\mathbb{R}^N,$$ where $B_{\\varepsilon}(x)$ is the open $\\varepsilon$-ball of centre $x$ and radius $\\varepsilon$. Using the critical point theory combining to the fractional Hardy inequality, we show that the problem $(P_+)$ admits at least two distinct nontrivial weak solutions. For the problem $(P_-),$ we use the concentration-compactness principle for fractional Sobolev spaces to give a weak lower semicontinuity result and prove that problem $(P_-)$ admits at least one non-trivial weak solution.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-01-29T20:26:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "singular elliptic problem",
      "fractional p-laplacian",
      "existence results",
      "weak lower semicontinuity result",
      "distinct nontrivial weak solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}