{
  "id": "1605.08206",
  "version": "v1",
  "published": "2016-05-26T09:32:01.000Z",
  "updated": "2016-05-26T09:32:01.000Z",
  "title": "Eigenvalue problem for a p-Laplacian equation with trapping potentials",
  "authors": [
    "Long-Jiang Gu",
    "Xiaoyu Zeng",
    "Huan-Song Zhou"
  ],
  "comment": "21pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Consider the following eigenvalue problem of p-Laplacian equation \\begin{equation}\\label{P} -\\Delta_{p}u+V(x)|u|^{p-2}u=\\mu|u|^{p-2}u+a| u|^{s-2}u, x\\in \\mathbb{R}^{n}, \\tag{P} \\end{equation} where $a\\geq0$, $p\\in (1,n)$ and $\\mu\\in\\mathbb{R}$. $V(x)$ is a trapping type potential, e.g., $\\inf\\limits_{x \\in \\mathbb{R}^n}V(x)< \\lim\\limits_{|x|\\rightarrow+\\infty }V(x)$. By using constrained variational methods, we proved that there is $a^*>0$, which can be given explicitly, such that problem (\\ref{P}) has a ground state $u$ with $\\|u\\|_{L^p}=1$ for some $\\mu \\in \\mathbb{R}$ and all $a\\in [0,a^*)$, but (\\ref{P}) has no this kind of ground state if $a\\geq a^*$. Furthermore, by establishing some delicate energy estimates we show that the global maximum point of the ground states of problem (\\ref{P}) approach to one of the global minima of $V(x)$ and blow up if $a\\nearrow a^*$. The optimal rate of blowup is obtained for $V(x)$ being a polynomial type potential.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-05-26T09:32:01.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35P30",
      "47J10",
      "35J92"
    ],
    "keywords": [
      "p-laplacian equation",
      "eigenvalue problem",
      "trapping potentials",
      "ground state",
      "polynomial type potential"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}