{
  "id": "0801.3379",
  "version": "v1",
  "published": "2008-01-22T14:35:50.000Z",
  "updated": "2008-01-22T14:35:50.000Z",
  "title": "Saddle-shaped solutions of bistable diffusion equations in all of $\\mathbb{R}^{2m}$",
  "authors": [
    "Xavier Cabre",
    "Joana Terra"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the existence and instability properties of saddle-shaped solutions of the semilinear elliptic equation $-\\Delta u = f(u)$ in the whole $\\R^{2m}$, where $f$ is of bistable type. It is known that in dimension $2m=2$ there exists a saddle-shaped solution. This is a solution which changes sign in $\\R^2$ and vanishes only on $\\{|x_1|=|x_2|\\}$. It is also known that this solution is unstable. In this article we prove the existence of saddle-shaped solutions in every even dimension, as well as their instability in the case of dimension $2m=4$. More precisely, our main result establishes that if $2m=4$, every solution vanishing on the Simons cone $\\{(x^1,x^2)\\in\\R^m\\times\\R^m : |x^1|=|x^2|\\}$ is unstable outside of every compact set and, as a consequence, has infinite Morse index. These results are relevant in connection with a conjecture of De Giorgi extensively studied in recent years and for which the existence of a counter-example in high dimensions is still an open problem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-01-22T14:35:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "saddle-shaped solution",
      "bistable diffusion equations",
      "semilinear elliptic equation",
      "main result establishes",
      "infinite morse index"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0801.3379C"
    }
  }
}