{
  "id": "2108.01414",
  "version": "v1",
  "published": "2021-08-03T11:13:51.000Z",
  "updated": "2021-08-03T11:13:51.000Z",
  "title": "Calculus of variations on locally finite graphs",
  "authors": [
    "Yong Lin",
    "Yunyan Yang"
  ],
  "comment": "19 pages",
  "categories": [
    "math.AP",
    "math.CO"
  ],
  "abstract": "Let $G=(V,E)$ be a locally finite graph. Firstly, using calculus of variations, including a direct method of variation and the mountain-pass theory, we get sequences of solutions to several local equations on $G$ (the Schr\\\"odinger equation, the mean field equation, and the Yamabe equation). Secondly, we derive uniform estimates for those local solution sequences. Finally, we obtain global solutions by extracting convergent sequence of solutions. Our method can be described as a variational method from local to global.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-03T11:13:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "35R02",
      "34B45"
    ],
    "keywords": [
      "locally finite graph",
      "mean field equation",
      "local solution sequences",
      "local equations",
      "direct method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}