{
  "id": "2411.01276",
  "version": "v1",
  "published": "2024-11-02T15:04:07.000Z",
  "updated": "2024-11-02T15:04:07.000Z",
  "title": "Nonlinear eigenvalue problems for a biharmonic operator in Orlicz-Sobolev spaces",
  "authors": [
    "Pablo Ochoa",
    "Analía Silva"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we introduce a new higher-order Laplacian operator in the framework of Orlicz-Sobolev spaces, the biharmonic g-Laplacian $$\\Delta_g^2 u:=\\Delta \\left(\\dfrac{g(|\\Delta u|)}{|\\Delta u|} \\Delta u\\right),$$ where $g=G'$, with $G$ an N-function. This operator is a generalization of the so called bi-harmonic Laplacian $\\Delta^2$. Here, we also established basic functional properties of $\\Delta_g^2$, which can be applied to existence results. Afterwards, we study the eigenvalues of $\\Delta_g^2$, which depend on normalisation conditions, due to the lack of homogeneity of the operator. Finally, we study different nonlinear eigenvalue problems associated to $\\Delta_g^2$ and we show regimes where the corresponding spectrum concentrate at $0$, $\\infty$ or coincide with $(0, \\infty)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-11-02T15:04:07.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "46E30",
      "35P30",
      "35D30"
    ],
    "keywords": [
      "nonlinear eigenvalue problems",
      "orlicz-sobolev spaces",
      "biharmonic operator",
      "higher-order laplacian operator",
      "established basic functional properties"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}