{
  "id": "1607.04548",
  "version": "v1",
  "published": "2016-07-15T15:15:30.000Z",
  "updated": "2016-07-15T15:15:30.000Z",
  "title": "Existence of positive solutions to some nonlinear equations on locally finite graphs",
  "authors": [
    "Alexander Grigor'yan",
    "Yong Lin",
    "Yunyan Yang"
  ],
  "comment": "15 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "Let $G=(V,E)$ be a locally finite graph, whose measure $\\mu(x)$ have positive lower bound, and $\\Delta$ be the usual graph Laplacian. Applying the mountain-pass theorem due to Ambrosetti-Rabinowitz, we establish existence results for some nonlinear equations, namely $\\Delta u+hu=f(x,u)$, $x\\in V$. In particular, we prove that if $h$ and $f$ satisfy certain assumptions, then the above mentioned equation has strictly positive solutions. Also, we consider existence of positive solutions of the perturbed equation $\\Delta u+hu=f(x,u)+\\epsilon g$. Similar problems have been extensively studied on the Euclidean space as well as on Riemannian manifolds.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-15T15:15:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "34B45",
      "35A15",
      "58E30"
    ],
    "keywords": [
      "locally finite graph",
      "positive solutions",
      "nonlinear equations",
      "usual graph laplacian",
      "positive lower bound"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}