{
  "id": "1811.03143",
  "version": "v1",
  "published": "2018-11-07T21:03:07.000Z",
  "updated": "2018-11-07T21:03:07.000Z",
  "title": "Abundance of entire solutions to nonlinear elliptic equations by the variational method",
  "authors": [
    "L. M. Lerman",
    "P. E. Naryshkin",
    "A. I. Nazarov"
  ],
  "comment": "30 pages, 28 figures",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study entire bounded solutions to the equation $\\Delta u - u + u^3 = 0$ in $\\mathbb R^2$. Our approach is purely variational and is based on concentration arguments and symmetry considerations. This method allows us to construct in a unified way several types of solutions with various symmetries (radial, breather type, rectangular, triangular, hexagonal, etc.), both positive and sign-changing. The method is also applicable for more general equations in any dimension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-07T21:03:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J20",
      "35J91",
      "35J92"
    ],
    "keywords": [
      "nonlinear elliptic equations",
      "entire solutions",
      "variational method",
      "study entire bounded solutions",
      "general equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}