{
  "id": "1807.04940",
  "version": "v1",
  "published": "2018-07-13T07:02:10.000Z",
  "updated": "2018-07-13T07:02:10.000Z",
  "title": "Sharp existence and nonexistence results for an elliptic equation associated with Caffarelli-Kohn-Nirenberg inequalities",
  "authors": [
    "John Villavert"
  ],
  "comment": "25 pages; submitted; comments are welcome",
  "categories": [
    "math.AP"
  ],
  "abstract": "This article establishes sharp existence and Liouville type theorems for the following nonlinear elliptic equation with singular coefficients, \\begin{equation*} -div (|x|^{a} D u ) = |x|^{b}u^{p}, ~ u > 0,\\, \\mbox{ in } \\mathbb{R}^N, \\end{equation*} where $N \\geq 3$, $p > 1$, and $b > a - 2 > -N$. In certain cases, this recovers the Euler-Lagrange equations connected with finding the best constant in Sobolev and Caffarelli-Kohn-Nirenberg inequalities. The first main result indicates that regular solutions exist if and only if the exponent $p$ is either critical or supercritical, i.e., either \\begin{equation*} p = \\frac{N + 2 + 2b - a}{N - 2 + a} ~ \\mbox{ or } ~ p > \\frac{N + 2 + 2b - a}{N - 2 + a}, \\mbox{ respectively}. \\end{equation*} Similarly, the second main result provides necessary and sufficient conditions for the existence of finite energy solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-13T07:02:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B33",
      "35B53",
      "35J15",
      "35J75",
      "35B40",
      "35B65"
    ],
    "keywords": [
      "caffarelli-kohn-nirenberg inequalities",
      "nonexistence results",
      "article establishes sharp existence",
      "liouville type theorems",
      "second main result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}