{
  "id": "2110.14640",
  "version": "v2",
  "published": "2021-10-27T16:37:24.000Z",
  "updated": "2022-03-02T00:05:38.000Z",
  "title": "System with weights and with critical Sobolev exponent",
  "authors": [
    "Asma Benhamida",
    "Rejeb Hadiji"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we investigate the minimization problem: $$ \\inf_{ \\displaystyle{\\begin{array}{lll} u \\in H_0^1(\\Omega), v \\in H_0^1(\\Omega),\\\\ \\quad \\| u \\|_{L^{q}} =1, \\quad \\| v \\|_{L^{q}} = 1 \\end{array}}} \\left[ \\frac{1}{2} \\int_{\\Omega} a(x) \\vert \\nabla u \\vert^2dx + \\displaystyle{ \\frac{1}{2} \\int_{\\Omega} b(x) \\vert \\nabla v \\vert^2dx } - \\lambda \\displaystyle{\\int_{\\Omega} uv dx} \\right] $$ with $q=\\frac{2N}{N-2}$, $ N \\geq 4$, $a$ and $ b $ are two continuous positive weights. We show the existence of solutions of the previous minimizing problem under some conditions on $a$, $b$, the dimension of space and the parameter $\\lambda$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-03-02T00:05:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35A01",
      "35A15",
      "35J57",
      "35J62"
    ],
    "keywords": [
      "critical sobolev exponent",
      "minimization problem",
      "continuous positive weights",
      "minimizing problem",
      "conditions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}