{
  "id": "2310.08264",
  "version": "v1",
  "published": "2023-10-12T12:11:59.000Z",
  "updated": "2023-10-12T12:11:59.000Z",
  "title": "Classification of solutions of higher order critical Choquard equation",
  "authors": [
    "Genggeng Huang",
    "Yating Niu"
  ],
  "comment": "31 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we classify the solutions of the following critical Choquard equation \\[ (-\\Delta)^{\\frac{n}{2}} u(x) = \\int_{\\mathbb{R}^n} \\frac{e^{\\frac{2n- \\mu}{2}u(y)}}{|x-y|^{\\mu}}dy e^{\\frac{2n- \\mu}{2}u(x)}, \\ \\text{in} \\ \\mathbb{R}^n, \\] where $ 0<\\mu < n$, $ n\\ge 2$. Suppose $ u(x) = o(|x|^2) \\ \\text{at} \\ \\infty $ for $ n \\geq 3$ and satisfies \\[ \\int_{\\mathbb{R}^n}e^{\\frac{2n- \\mu}{2}u(y)} dy < \\infty, \\ \\int_{\\mathbb{R}^n}\\int_{\\mathbb{R}^n}\\frac{e^{\\frac{2n- \\mu}{2}u(y)}}{|x-y|^{\\mu}} e^{\\frac{2n- \\mu}{2}u(x)} dy dx < \\infty. \\] By using the method of moving spheres, we show that the solutions have the following form \\[ u(x)= \\ln \\frac{C_1(\\varepsilon)}{|x-x_0|^2 + \\varepsilon^2}. \\]",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-12T12:11:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J30",
      "35R11",
      "31B10"
    ],
    "keywords": [
      "higher order critical choquard equation",
      "classification",
      "dy dx",
      "moving spheres"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}