{
  "id": "1206.3196",
  "version": "v2",
  "published": "2012-06-14T17:50:09.000Z",
  "updated": "2012-06-18T19:55:49.000Z",
  "title": "A concentration phenomenon for semilinear elliptic equations",
  "authors": [
    "Nils Ackermann",
    "Andrzej Szulkin"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "For a domain $\\Omega\\subset\\dR^N$ we consider the equation $ -\\Delta u + V(x)u = Q_n(x)\\abs{u}^{p-2}u$ with zero Dirichlet boundary conditions and $p\\in(2,2^*)$. Here $V\\ge 0$ and $Q_n$ are bounded functions that are positive in a region contained in $\\Omega$ and negative outside, and such that the sets $\\{Q_n>0\\}$ shrink to a point $x_0\\in\\Omega$ as $n\\to\\infty$. We show that if $u_n$ is a nontrivial solution corresponding to $Q_n$, then the sequence $(u_n)$ concentrates at $x_0$ with respect to the $H^1$ and certain $L^q$-norms. We also show that if the sets $\\{Q_n>0\\}$ shrink to two points and $u_n$ are ground state solutions, then they concentrate at one of these points.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-06-18T19:55:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J61",
      "35Q55",
      "35Q60",
      "35B30",
      "35J20"
    ],
    "keywords": [
      "semilinear elliptic equations",
      "concentration phenomenon",
      "zero dirichlet boundary conditions",
      "ground state solutions",
      "nontrivial solution"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s00205-012-0589-1",
      "journal": "Archive for Rational Mechanics and Analysis",
      "year": 2013,
      "month": "Mar",
      "volume": 207,
      "number": 3,
      "pages": 1075
    },
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013ArRMA.207.1075A"
    }
  }
}