{
  "id": "2008.07398",
  "version": "v1",
  "published": "2020-08-17T15:13:28.000Z",
  "updated": "2020-08-17T15:13:28.000Z",
  "title": "Regularity results for Choquard equations involving fractional $p$-Laplacian",
  "authors": [
    "Reshmi Biswas",
    "Sweta Tiwari"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this article first we address the regularity of weak solution for a class of $p$-fractional Choquard equations: \\begin{equation*} \\;\\;\\; \\left.\\begin{array}{rl} (-\\Delta)_p^su &= \\left(\\displaystyle\\int_\\Omega\\frac{F(y,u)}{|x-y|^{\\mu}}dy\\right)f(x,u),\\hspace{5mm}x\\in \\Omega,\\\\ u &=0,\\hspace{50mm}x\\in \\mathbb R^N\\setminus \\Omega, \\end{array} \\right\\} \\end{equation*} where $\\Omega\\subset\\mathbb R^N$ is a smooth bounded domain, $1<p<\\infty$ and $0<s<1$ such that $sp<N,$ $\\mu<\\min\\{N,2sp\\}$ and $f:\\Omega\\times\\mathbb R\\to\\mathbb R$ is a continuous function with at most critical growth condition (in the sense of Hardy-Littlewood-Sobolev inequality) and $F$ is its primitive. Next for $p\\geq2$ we discuss the Sobolev versus H\\\"{o}lder minimizers of the energy functional $J$ associated to the above problem, and using that we establish the existence of the local minimizer of $J$ in the fractional Sobolev space $W_0^{s,p}(\\Omega).$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-08-17T15:13:28.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "regularity results",
      "fractional choquard equations",
      "fractional sobolev space",
      "article first",
      "weak solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}