{
  "id": "1905.11431",
  "version": "v1",
  "published": "2019-05-27T18:07:09.000Z",
  "updated": "2019-05-27T18:07:09.000Z",
  "title": "Semilinear integro-differential equations, II: one-dimensional and saddle-shaped solutions to the Allen-Cahn equation",
  "authors": [
    "Juan-Carlos Felipe-Navarro",
    "Tomás Sanz-Perela"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "This paper, which is the follow-up to part I, concerns saddle-shaped solutions to the semilinear equation $L_K u = f(u)$ in $\\mathbb{R}^{2m}$, where $L_K$ is a linear elliptic integro-differential operator with a radially symmetric kernel and $f$ is of Allen-Cahn type. Saddle-shaped solutions are doubly radial, odd with respect to the Simons cone $\\{(x', x'') \\in \\mathbb{R}^m \\times \\mathbb{R}^m \\, : \\, |x'| = |x''|\\}$, and vanish only in this set. Following the setting established in part I for doubly radial odd functions, we show existence, asymptotic behavior, and uniqueness of the saddle-shaped solution. For this, we prove, among others, a Liouville type result, the one-dimensional symmetry of positive solutions to semilinear problems in a half-space, and maximum principles in \"narrow\" sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-05-27T18:07:09.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35B08",
      "35B06",
      "47G20",
      "35B50",
      "35B40"
    ],
    "keywords": [
      "saddle-shaped solution",
      "semilinear integro-differential equations",
      "allen-cahn equation",
      "linear elliptic integro-differential operator",
      "liouville type result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}