{
  "id": "math/0202243",
  "version": "v1",
  "published": "2002-02-23T07:45:13.000Z",
  "updated": "2002-02-23T07:45:13.000Z",
  "title": "Combining solutions of semilinear partial differential equations in R^n with critical exponent",
  "authors": [
    "Man Chun Leung"
  ],
  "comment": "35 pages",
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "Let $u_1$ and $u_2$ be two different positive smooth solutions of the equation $\\Delta u + n (n - 2) u^{{n + 2}\\over {n - 2}} = 0$ in $R^n (n \\ge 3).$ By a result of Gidas, Ni and Nirenberg, $u_1$ and $u_2$ are radially symmetric above the points $\\xi_1$ and $\\xi_2$, respectively. Let $u$ be a positive $C^2$-function on $R^n$ such that $u = u_1$ in $\\Omega_1$ and $u = u_2$ in $\\Omega_2$, where $\\Omega_1$ and $\\Omega_2$ are disjoint non-empty open domains in ${\\R}^n$. $u$ satisfies the equation $\\Delta u + n (n - 2) K u^{{n + 2}\\over {n - 2}} = 0$ in $R^n.$ By the same result of Gidas, Ni and Nirenberg, $K \\not\\equiv 1$ in $R^n$. In this paper we discuss lower bounds on $\\displaystyle{\\sup_{\\R^n} |K - 1|} .$ Relation with decay estimates at the isolated singularity via the Kelvin transform is also considered.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-02-23T07:45:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "53C21"
    ],
    "keywords": [
      "semilinear partial differential equations",
      "critical exponent",
      "combining solutions",
      "disjoint non-empty open domains",
      "positive smooth solutions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......2243L"
    }
  }
}