{
  "id": "2403.08146",
  "version": "v1",
  "published": "2024-03-13T00:17:48.000Z",
  "updated": "2024-03-13T00:17:48.000Z",
  "title": "Nodal solutions to Paneitz-type equations",
  "authors": [
    "Jurgen Julio-Batalla",
    "Jimmy Petean"
  ],
  "categories": [
    "math.AP",
    "math.DG"
  ],
  "abstract": "On a closed Riemannian manifold $(M^n ,g)$ with a proper isoparametric function $f$ we consider the equation $\\Delta^2 u -\\alpha \\Delta u +\\beta u = u^q$, where $\\alpha$ and $\\beta$ are positive constants satisfying that $\\alpha^2 \\geq 4 \\beta$. We let ${\\bf m}$ be the minimum of the dimensions of the focal varieties of $f$ and $q_f = \\frac{n-{\\bf m}+4}{n-{\\bf m}-4}$, $q_f = \\infty$ if $n\\leq {\\bf m}+4$. We prove the existence of infinitely many nodal solutions of the equation assuming that $1<q<q_f$. The solutions are $f$-invariant. To obtain the result, first we prove a $C^0-$estimate for positive $f$-invariant solutions of the equation. Then we prove the existence of mountain pass solutions with arbitrarily large energy.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-13T00:17:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nodal solutions",
      "paneitz-type equations",
      "proper isoparametric function",
      "mountain pass solutions",
      "closed riemannian manifold"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}