{
  "id": "2203.08482",
  "version": "v1",
  "published": "2022-03-16T09:18:17.000Z",
  "updated": "2022-03-16T09:18:17.000Z",
  "title": "Multiple solutions for Schrödinger equations on Riemannian manifolds via $\\nabla$-theorems",
  "authors": [
    "Luigi Appolloni",
    "Giovanni Molica Bisci",
    "Simone Secchi"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "We consider a smooth, complete and non-compact Riemannian manifold $(\\mathcal{M},g)$ of dimension $d \\geq 3$, and we look for positive solutions to the semilinear elliptic equation $$ -\\Delta_g w + V w = \\alpha f(w) + \\lambda w \\quad\\hbox{in $\\mathcal{M}$}. $$ The potential $V \\colon \\mathcal{M} \\to \\mathbb{R}$ is a continuous function which is coercive in a suitable sense, while the nonlinearity $f$ has a subcritical growth in the sense of Sobolev embeddings. By means of $\\nabla$-Theorems introduced by Marino and Saccon, we prove that at least three solution exists as soon as the parameter $\\lambda$ is sufficiently close to an eigenvalue of the operator $-\\Delta_g$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-03-16T09:18:17.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "schrödinger equations",
      "multiple solutions",
      "non-compact riemannian manifold",
      "semilinear elliptic equation",
      "sobolev embeddings"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}