{
  "id": "1202.3980",
  "version": "v4",
  "published": "2012-02-17T17:51:21.000Z",
  "updated": "2012-08-15T15:50:36.000Z",
  "title": "On the resonant Lane-Emden problem for the p-Laplacian",
  "authors": [
    "Grey Ercole"
  ],
  "comment": "The paper was improved and completely reformulated. The main result now is valid for any family of positive solutions in the super-linear case (not necessarily ground state)",
  "doi": "10.1142/S0219199713500338",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the positive solutions of the Lane-Emden equation $-\\Delta_{p}u=\\lambda_{p}|u|^{q-2}u$ in $\\Omega$ with homogeneous Dirichlet boundary conditions, where $\\Omega\\subset\\mathbb{R}^{N}$ is a bounded and smooth domain, $N\\geq2,$ $\\lambda_{p}$ is the first eigenvalue of the $p$-Laplacian operator $\\Delta_{p}$ and $q$ is close to $p>1.$ We prove that any family of positive solutions of this problem converges in $C^{1}(\\bar{\\Omega})$ to the function $\\theta_{p}e_{p}$ when $q\\rightarrow p,$ where $e_{p}$ is the positive and $L^{\\infty}$-normalized first eigenfunction of the $p$-Laplacian and $\\theta_{p}:=\\exp(|e_{p}|_{L^{p}(\\Omega)}^{-p}\\int_{\\Omega}e_{p}% ^{p}|\\ln e_{p}|dx).$ A consequence of this result is that the best constant of the immersion $W_{0}^{1,p}(\\Omega)\\hookrightarrow L^{q}(\\Omega)$ is differentiable at $q=p.$ Previous results on the asymptotic behavior (as $q\\rightarrow p$) of the positive solutions of the non-resonant Lane-Emden problem (i.e. with $\\lambda_{p}$ replaced by a positive $\\lambda\\neq\\lambda_{p}$) are also generalized to the space $C^{1}% (\\bar{\\Omega})$ and to arbitrary families of these solutions. Moreover, if $u_{\\lambda,q}$ denotes a solution of the non-resonant problem for an arbitrarily fixed $\\lambda>0,$ we show how to obtain the first eigenpair of the $p$-Laplacian as the limit in $C^{1}(\\bar{\\Omega}),$ when $q\\rightarrow p$, of a suitable scaling of the pair $(\\lambda,u_{\\lambda,q}).$ For computational purposes the advantage of this approach is that $\\lambda$ does not need to be close to $\\lambda_{p}.$ Finally, an explicit estimate involving $L^{\\infty}$ and $L^{1}$ norms of $u_{\\lambda,q}$ is also deduced using set level techniques.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2012-08-15T15:50:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B35",
      "35B40",
      "35B44",
      "35J92",
      "65N25"
    ],
    "keywords": [
      "positive solutions",
      "p-laplacian",
      "set level techniques",
      "homogeneous dirichlet boundary conditions",
      "non-resonant lane-emden problem"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.3980E"
    }
  }
}