{
  "id": "2102.12212",
  "version": "v1",
  "published": "2021-02-24T11:12:00.000Z",
  "updated": "2021-02-24T11:12:00.000Z",
  "title": "Influence of an $L^p$-perturbation on Hardy-Sobolev inequality with singularity a curve",
  "authors": [
    "El Hadji Abdoulaye Thiam",
    "Idowu Esther IJaodoro"
  ],
  "comment": "arXiv admin note: substantial text overlap with arXiv:1702.02202",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider a bounded domain $\\Omega$ of $\\mathbb{R}^N$, $N\\ge3$, $h$ and $b$ continuous functions on $\\Omega$. Let $\\Gamma$ be a closed curve contained in $\\Omega$. We study existence of positive solutions $u \\in H^1_0\\left(\\Omega\\right)$ to the perturbed Hardy-Sobolev equation: $$ -\\Delta u+h u+bu^{1+\\delta}=\\rho^{-\\sigma}_\\Gamma u^{2^*_\\sigma-1} \\qquad \\textrm{ in } \\Omega, $$ where $2^*_\\sigma:=\\frac{2(N-\\sigma)}{N-2}$ is the critical Hardy-Sobolev exponent, $\\sigma\\in [0,2)$, $0< \\delta<\\frac{4}{N-2}$ and $\\rho_\\Gamma$ is the distance function to $\\Gamma$. We show that the existence of minimizers does not depend on the local geometry of $\\Gamma$ nor on the potential $h$. For $N=3$, the existence of ground-state solution may depends on the trace of the regular part of the Green function of $-\\Delta+h$ and or on $b$. This is due to the perturbative term of order ${1+\\delta}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-24T11:12:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hardy-sobolev inequality",
      "singularity",
      "perturbation",
      "ground-state solution",
      "local geometry"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}