{
  "id": "2209.14418",
  "version": "v1",
  "published": "2022-09-28T21:06:28.000Z",
  "updated": "2022-09-28T21:06:28.000Z",
  "title": "The method of the energy function and applications",
  "authors": [
    "Claudianor O. Alves",
    "Tiago L. Coelho",
    "João R. Santos Júnior"
  ],
  "comment": "28 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "In this work, we establish a new method to find critical points of differentiable functionals defined in Banach spaces which belong to a suitable class ($\\mathcal{J}$) of functionals. Once given a functional $J$ in the class ($\\mathcal{J}$), the central idea of the referred method consists in defining a real function $\\zeta$ of a real variable, called {\\it energy function}, which is naturally associated to $J$ in the sense that the existence of real critical points for $\\zeta$ guarantees the existence of critical points for the functional $J$. As a consequence, we are able to solve some variational elliptic problems, whose associated energy functional belongs to ($\\mathcal{J}$) and provide a version of the mountain pass theorem for functionals in the class ($\\mathcal{J}$) that allows us to obtain mountain pass solutions without the so-called Ambrosetti-Rabinowitz condition.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-09-28T21:06:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35J20",
      "35J25",
      "35Q74"
    ],
    "keywords": [
      "critical points",
      "applications",
      "variational elliptic problems",
      "mountain pass solutions",
      "mountain pass theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}