{
  "id": "1709.04888",
  "version": "v1",
  "published": "2017-09-14T17:24:52.000Z",
  "updated": "2017-09-14T17:24:52.000Z",
  "title": "Sign-Changing Solutions for Critical Equations with Hardy Potential",
  "authors": [
    "Pierpaolo Esposito",
    "Nassif Ghoussoub",
    "Angela Pistoia",
    "Giusi Vaira"
  ],
  "comment": "41 pages, Updated version - if any - can be downloaded at http://www.birs.ca/~nassif/",
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the following perturbed critical Dirichlet problem involving the Hardy-Schr\\\"odinger operator on a smooth bounded domain $\\Omega \\subset \\mathbb{R}^N$, $N\\geq 3$, with $0 \\in \\Omega$: $$ \\left\\{ \\begin{array}{ll}-\\Delta u-\\gamma \\frac{u}{|x|^2}-\\epsilon u=|u|^{\\frac{4}{N-2}}u &\\hbox{in }\\Omega u=0 & \\hbox{on }\\partial \\Omega, \\end{array}\\right. $$ when $\\epsilon>0$ is small and $\\gamma< {(N-2)^2\\over4}$. Setting $ \\gamma_j= \\frac{(N-2)^2}{4}\\left(1-\\frac{j(N-2+j)}{N-1}\\right)\\in(-\\infty,0]$ for $j \\in \\mathbb{N},$ we show that if $\\gamma\\leq \\frac{(N-2)^2}{4}-1$ and $\\gamma \\neq \\gamma_j$ for any $j$, then for small $\\epsilon$, the above equation has a positive --non variational-- solution that develops a bubble at the origin. If moreover $\\gamma<\\frac{(N-2)^2}{4}-4,$ then for any integer $k \\geq 2$, the equation has for small enough $\\epsilon$, a sign-changing solution that develops into a superposition of $k$ bubbles with alternating sign centered at the origin. The above result is optimal in the radial case, where the condition that $\\gamma\\neq \\gamma_j$ is not necessary. Indeed, it is known that, if $\\gamma > \\frac{(N-2)^2}{4}-1$ and $\\Omega$ is a ball $B$, then there is no radial positive solution for $\\epsilon>0$ small. We complete the picture here by showing that, if $\\gamma\\geq \\frac{(N-2)^2}{4}-4$, then the above problem has no radial sign-changing solutions for $\\epsilon>0$ small. These results recover and improve what is known in the non-singular case, i.e., when $\\gamma=0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-14T17:24:52.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hardy potential",
      "critical equations",
      "radial sign-changing solutions",
      "radial positive solution",
      "smooth bounded domain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 41,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}