{
  "id": "0907.3008",
  "version": "v1",
  "published": "2009-07-17T14:21:13.000Z",
  "updated": "2009-07-17T14:21:13.000Z",
  "title": "Qualitative properties of saddle-shaped solutions to bistable diffusion equations",
  "authors": [
    "Xavier Cabre",
    "Joana Terra"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider the elliptic equation $-\\Delta u = f(u)$ in the whole $\\R^{2m}$, where $f$ is of bistable type. It is known that there exists a saddle-shaped solution in $\\R^{2m}$. This is a solution which changes sign in $\\R^{2m}$ and vanishes only on the Simons cone ${\\mathcal C}=\\{(x^1,x^2)\\in\\R^m\\times\\R^m: |x^1|=|x^2|\\}$. It is also known that these solutions are unstable in dimensions 2 and 4. In this article we establish that when $2m=6$ every saddle-shaped solution is unstable outside of every compact set and, as a consequence has infinite Morse index. For this we establish the asymptotic behavior of saddle-shaped solutions at infinity. Moreover we prove the existence of a minimal and a maximal saddle-shaped solutions and derive monotonicity properties for the maximal solution. These results are relevant in connection with a conjecture of De Giorgi on 1D symmetry of certain solutions. Saddle-shaped solutions are the simplest candidates, besides 1D solutions, to be global minimizers in high dimensions, a property not yet established.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-07-17T14:21:13.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "bistable diffusion equations",
      "qualitative properties",
      "infinite morse index",
      "1d solutions",
      "high dimensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.3008C"
    }
  }
}