{
  "id": "1507.03056",
  "version": "v1",
  "published": "2015-07-11T02:24:31.000Z",
  "updated": "2015-07-11T02:24:31.000Z",
  "title": "On a biharmonic equations with steep potential well and indefinite potential",
  "authors": [
    "Yisheng Huang",
    "Zeng Liu",
    "Yuanze Wu"
  ],
  "comment": "18 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we study the following biharmonic equations:% $$ \\left\\{\\aligned&\\Delta^2u-a_0\\Delta u+(\\lambda b(x)+b_0)u=f(u)&\\text{ in }\\bbr^N,\\\\% &u\\in\\h,\\endaligned\\right.\\eqno{(\\mathcal{P}_{\\lambda})}% $$ where $N\\geq3$, $a_0,b_0\\in\\bbr$ are two constants, $\\lambda>0$ is a parameter, $b(x)\\geq0$ is a potential well and $f(t)\\in C(\\bbr)$ is subcritical and superlinear or asymptotically linear at infinity. By the Gagliardo-Nirenberg inequality, we make some observations on the operator $\\Delta^2-a_0\\Delta+\\lambda b(x)+b_0$ in $\\h$. Based on these observations, we give a new variational setting to $(\\mathcal{P}_{\\lambda})$ for $a_0<0$. With this new variational setting in hands, we establish some new existence results of the nontrivial solutions to $(\\mathcal{P}_{\\lambda})$ for all $a_0, b_0\\in\\bbr$ with $\\lambda$ sufficiently large by the variational method. The concentration behavior of the nontrivial solutions as $\\lambda\\to+\\infty$ is also obtained. It is worth to point out that it seems to be the first time that the nontrivial solution of $(\\mathcal{P}_{\\lambda})$ is obtained in the case of $a_0<0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-07-11T02:24:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "steep potential",
      "indefinite potential",
      "biharmonic equations",
      "nontrivial solution",
      "gagliardo-nirenberg inequality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}