{
  "id": "2305.19781",
  "version": "v1",
  "published": "2023-05-31T12:17:06.000Z",
  "updated": "2023-05-31T12:17:06.000Z",
  "title": "On a class of elliptic equations with Critical Perturbations in the hyperbolic space",
  "authors": [
    "Debdip Ganguly",
    "Diksha Gupta",
    "K. Sreenadh"
  ],
  "comment": "26 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We study the existence and non-existence of positive solutions for the following class of nonlinear elliptic problems in the hyperbolic space $$ -\\Delta_{\\mathbb{B}^N} u-\\lambda u=a(x)u^{p-1} \\, + \\, \\varepsilon u^{2^*-1} \\,\\;\\;\\text{in}\\;\\mathbb{B}^{N}, \\quad u \\in H^{1}{(\\mathbb{B}^{N})}, $$ where $\\mathbb{B}^N$ denotes the hyperbolic space, $2<p<2^*:=\\frac{2N}{N-2}$, if $N \\geqslant 3; 2<p<+\\infty$, if $N = 2,\\;\\lambda < \\frac{(N-1)^2}{4}$, and $0< a\\in L^\\infty(\\mathbb{B}^N).$ We first prove the existence of a positive radially symmetric ground-state solution for $a(x) \\equiv 1.$ Next, we prove that for $a(x) \\geq 1$, there exists a ground-state solution for $\\varepsilon$ small. For proof, we employ ``conformal change of metric\" which allows us to transform the original equation into a singular equation in a ball in $\\mathbb R^N$. Then by carefully analysing the energy level using blow-up arguments, we prove the existence of a ground-state solution. Finally, the case $a(x) \\leq 1$ is considered where we first show that there is no ground-state solution, and prove the existence of a \\it bound-state solution \\rm (high energy solution) for $\\varepsilon$ small. We employ variational arguments in the spirit of Bahri-Li to prove the existence of high energy-bound-state solutions in the hyperbolic space.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-31T12:17:06.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hyperbolic space",
      "elliptic equations",
      "critical perturbations",
      "positive radially symmetric ground-state solution",
      "nonlinear elliptic problems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}