{
  "id": "math/0609302",
  "version": "v1",
  "published": "2006-09-11T17:00:14.000Z",
  "updated": "2006-09-11T17:00:14.000Z",
  "title": "On the Yamabe equation with rough potentials",
  "authors": [
    "Francesca Prinari",
    "Nicola Visciglia"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the existence of non--trivial solutions to the Yamabe equation: $$-\\Delta u+ a(x)= \\mu u|u|^\\frac4{n-2} \\hbox{} \\mu >0, x\\in \\Omega \\subset {\\mathbf R}^n \\hbox{with} n\\geq 4,$$ $$ u(x)=0 \\hbox{on} \\partial \\Omega$$ under weak regularity assumptions on the potential $a(x)$. More precisely in dimension $n\\geq 5$ we assume that: \\begin{enumerate} \\item $a(x)$ belongs to the Lorentz space $L^{\\frac n2, d}(\\Omega)$ for some $1\\leq d <\\infty$, \\item $a(x) \\leq M<\\infty \\hbox{a.e.} x\\in \\Omega$, \\item the set $\\{x\\in \\Omega|a(x)<0\\}$ has positive measure, \\item there exists $c>0$ such that $$\\int_\\Omega (|\\nabla u|^2 + a(x) |u|^2) \\hbox{} dx \\geq c\\int_\\Omega |\\nabla u|^2 \\hbox{} dx \\hbox{} \\forall u\\in H^1_0(\\Omega).$$ \\end{enumerate} \\noindent In dimension $n=4$ the hypothesis $(2)$ above is replaced by $$a(x)\\leq 0 \\hbox{} a.e. \\hbox{} x\\in \\Omega.$$",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-09-11T17:00:14.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "yamabe equation",
      "rough potentials",
      "weak regularity assumptions",
      "non-trivial solutions",
      "lorentz space"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......9302P"
    }
  }
}