{
  "id": "1407.3572",
  "version": "v1",
  "published": "2014-07-14T09:08:34.000Z",
  "updated": "2014-07-14T09:08:34.000Z",
  "title": "Moderate solutions of semilinear elliptic equations with Hardy potential",
  "authors": [
    "Moshe Marcus",
    "Phuoc-Tai Nguyen"
  ],
  "comment": "23 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $\\Omega$ be a bounded smooth domain in $\\mathbb{R}^N$. We study positive solutions of equation (E) $-L_\\mu u+ u^q = 0$ in $\\Omega$ where $L_\\mu=\\Delta + \\frac{\\mu}{\\delta^2}$, $0<\\mu$, $q>1$ and $\\delta(x)=\\mathrm{dist}\\,(x,\\partial\\Omega)$. A positive solution of (E) is moderate if it is dominated by an $L_\\mu$-harmonic function. If $\\mu<C_H(\\Omega)$ (the Hardy constant for $\\Omega$) every positive $L_\\mu$- harmonic functions can be represented in terms of a finite measure on $\\partial\\Omega$ via the Martin representation theorem. However the classical measure boundary trace of any such solution is zero. We introduce a notion of normalized boundary trace by which we obtain a complete classification of the positive moderate solutions of (E) in the subcritical case, $1<q<q_{\\mu,c}$. (The critical value depends only on $N$ and $\\mu$.) For $q\\geq q_{\\mu,c}$ there exists no moderate solution with an isolated singularity on the boundary. The normalized boundary trace and associated boundary value problems are also discussed in detail for the linear operator $L_\\mu$. These results form the basis for the study of the nonlinear problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-07-14T09:08:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J60",
      "35J75",
      "35J10"
    ],
    "keywords": [
      "semilinear elliptic equations",
      "hardy potential",
      "normalized boundary trace",
      "harmonic function",
      "positive solution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1407.3572M"
    }
  }
}