{
  "id": "1801.01544",
  "version": "v1",
  "published": "2018-01-04T20:57:27.000Z",
  "updated": "2018-01-04T20:57:27.000Z",
  "title": "Boundary value problem with measures for fractional elliptic equations involving source nonlinearities",
  "authors": [
    "Mousomi Bhakta",
    "Phuoc-Tai Nguyen"
  ],
  "comment": "27 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We are concerned with positive solutions of equation (E) $(-\\Delta)^s u=f(u)$ in a domain $\\Omega \\subset \\mathbb{R}^N$ ($N>2s$), where $s \\in (\\frac{1}{2},1)$ and $f\\in C^{\\alpha}_{loc}(\\mathbb{R})$ for some $\\alpha \\in(0,1)$. We establish a universal a priori estimate for positive solutions of (E), as well as for their gradients. Then for $C^2$ bounded domain $\\Omega$, we prove the existence of positive solutions of (E) with prescribed boundary value $\\rho \\nu$, where $\\rho>0$ and $\\nu$ is a positive Radon measure on $\\partial \\Omega$ with total mass $1$, and discuss regularity property of the solutions. When $f(u)=u^p$, we demonstrate that there exists a critical exponent $p_s:=\\frac{N+s}{N-s}$ in the following sense. If $p\\geq p_s$, the problem does not admit any positive solution with $\\nu$ being a Dirac mass. If $p\\in(1,p_s)$ there exits a threshold value $\\rho^*>0$ such that for $\\rho\\in (0, \\rho^*]$, the problem admits a positive solution and for $\\rho>\\rho^*$, no positive solution exists. We also show that, for $\\rho>0$ small enough, the problem admits at least two positive solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-04T20:57:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R06",
      "35R11",
      "35J66",
      "35J20"
    ],
    "keywords": [
      "positive solution",
      "fractional elliptic equations",
      "boundary value problem",
      "source nonlinearities",
      "problem admits"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}