{
  "id": "2306.12039",
  "version": "v1",
  "published": "2023-06-21T06:08:47.000Z",
  "updated": "2023-06-21T06:08:47.000Z",
  "title": "Classification of solutions to the anisotropic $N$-Liouville equation in $\\mathbb{R}^N$",
  "authors": [
    "Giulio Ciraolo",
    "Xiaoliang Li"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "Given $N\\geq 2$, we completely classify the solutions of the anisotropic $N$-Liouville equation $$-\\Delta_N^H\\,u=e^u \\quad\\text{in }\\mathbb{R}^N,$$ under the finite mass condition $\\int_{\\mathbb{R}^N} e^u\\,dx<+\\infty$. Here $\\Delta_N^H$ is the so-called Finsler $N$-Laplacian induced by a positively homogeneous function $H$. As a consequence for $N=2$, we give an affirmative answer to a conjecture made in [G. Wang and C. Xia, J. Differential Equations 252 (2012) 1668--1700].",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-21T06:08:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "liouville equation",
      "anisotropic",
      "classification",
      "finite mass condition",
      "differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}