{
  "id": "1510.03803",
  "version": "v1",
  "published": "2015-10-13T18:06:34.000Z",
  "updated": "2015-10-13T18:06:34.000Z",
  "title": "Semilinear elliptic equations with Hardy potential and subcritical source term",
  "authors": [
    "Phuoc-Tai Nguyen"
  ],
  "comment": "26 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\Omega$ be a smooth bounded domain in $\\mathbb{R}^N$ and $\\delta(x)=\\text{dist}\\,(x,\\partial \\Omega)$. Assume $\\mu>0$, $\\nu$ is a nonnegative finite measure on $\\partial \\Omega$ and $g \\in C(\\Omega \\times \\mathbb{R}_+)$. We study positive solutions of $$ (P)\\qquad -\\Delta u - \\frac{\\mu}{\\delta^2} u = g(x,u) \\text{ in } \\Omega, \\qquad \\text{tr}^*(u)=\\nu. $$ Here $\\text{tr}^*(u)$ denotes the \\textit{normalized boundary trace} of $u$ which was recently introduced by M. Marcus and P. T. Nguyen. We focus on the case $0<\\mu < C_H(\\Omega)$ (the Hardy constant for $\\Omega$) and provide some qualitative properties of solutions of (P). When $g(x,u)=u^q$ with $q>1$, we prove that there is a critical value $q^*$ (depending only on $N$, $\\mu$) for (P) in the sense that if $1<q<q^*$ then (P) admits a solution under a smallness assumption on $\\nu$, but if $q \\geq q^*$ this problem admits no solution with isolated boundary singularity. Existence result is then extended to a more general setting where $g$ is \\textit{subcritical}. We also investigate the case where the $g$ is linear or sublinear and give some existence results for (P).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-10-13T18:06:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "35J75",
      "35J10"
    ],
    "keywords": [
      "semilinear elliptic equations",
      "subcritical source term",
      "hardy potential",
      "existence result",
      "smooth bounded domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}