{
  "id": "2310.07447",
  "version": "v1",
  "published": "2023-10-11T12:50:16.000Z",
  "updated": "2023-10-11T12:50:16.000Z",
  "title": "Generalized solutions to semilinear elliptic equations with measure data",
  "authors": [
    "Tomasz Klimsiak"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We solve an open problem posed by H. Brezis, M. Marcus and A.C. Ponce in: Nonlinear elliptic equations with measures revisited. In: Mathematical Aspects of Nonlinear Dispersive Equations (J. Bourgain, C. Kenig, S. Klainerman, eds.), Annals of Mathematics Studies, 163 (2007). We prove that for any bounded Borel measure $\\mu$ on a smooth bounded domain $D\\subset\\mathbb R^d$ and non-increasing (with respect to the second variable) non-positive continuous function $f$ on $D\\times\\mathbb R$ the sequence of solutions to the semi-linear equation (P): $-\\Delta u=f(\\cdot,u)+\\rho_n\\ast\\mu$ ($\\rho_n$ is a mollifier) that is subject to homogeneous Dirichlet condition, converges to the function that solves (P) with $\\rho_n\\ast\\mu$ replaced by the reduced measure $\\mu^*$ (metric projection onto the space of good measures). We also provide a corresponding version of this result without non-positivity assumption on $f$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-11T12:50:16.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "semilinear elliptic equations",
      "measure data",
      "generalized solutions",
      "nonlinear elliptic equations",
      "non-positivity assumption"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}