{
  "id": "1902.04351",
  "version": "v1",
  "published": "2019-02-12T12:14:23.000Z",
  "updated": "2019-02-12T12:14:23.000Z",
  "title": "On Helmholtz equations and counterexamples to Strichartz estimates in hyperbolic space",
  "authors": [
    "Jean-Baptiste Casteras",
    "Rainer Mandel"
  ],
  "categories": [
    "math.AP"
  ],
  "abstract": "In this paper, we study nonlinear Helmholtz equations (NLH) $-\\Delta_{\\mathbb{H}^N} u - \\frac{(N-1)^2}{4} u -\\lambda^2 u = \\Gamma|u|^{p-2}u$ in $\\mathbb{H}^N$, $N\\geq 2$ where $\\Delta_{\\mathbb{H}^N}$ denotes the Laplace-Beltrami operator in the hyperbolic space $\\mathbb{H}^N$ and $\\Gamma\\in L^\\infty(\\mathbb{H}^N)$ is chosen suitably. Using fixed point and variational techniques, we find nontrivial solutions to (NLH) for all $\\lambda>0$ and $p>2$. The oscillatory behaviour and decay rates of radial solutions is analyzed, with possible extensions to Cartan-Hadamard manifolds and Damek-Ricci spaces. Our results rely on a new Limiting Absorption Principle for the Helmholtz operator in $\\mathbb{H}^N$. As a byproduct, we obtain simple counterexamples to certain Strichartz estimates.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-02-12T12:14:23.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hyperbolic space",
      "strichartz estimates",
      "study nonlinear helmholtz equations",
      "simple counterexamples",
      "helmholtz operator"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}