{
  "id": "2310.10577",
  "version": "v1",
  "published": "2023-10-16T16:53:37.000Z",
  "updated": "2023-10-16T16:53:37.000Z",
  "title": "Uniqueness and nondegeneracy of solutions to $(-Δ)^su+ u=u^p$ in $\\mathbb{R}^N$ and in balls",
  "authors": [
    "Mouhamed Moustapha Fall",
    "Tobias Weth"
  ],
  "comment": "15 pages",
  "categories": [
    "math.AP"
  ],
  "abstract": "We prove that positive solutions $u\\in H^s(\\mathbb{R}^N)$ to the equation $(-\\Delta )^s u+ u=u^p$ in $\\mathbb{R}^N$ are unique up to translations and nondegenerate, for all $s\\in (0,1)$, $N\\geq 1$ and $p>1$ is strictly smaller than the critical Sobolev exponent. This generalizes a result of Frank, Lenzmann and Silvestre \\cite{FLS}, where the same uniqueness and nondegeneracy is proved for solutions with Morse index 1. Letting $B$ be the unit centered ball and $\\lambda_1(B)$ the first Dirichlet eigenvalue of the fractional Laplacian $(-\\Delta )^s$, we also prove that positive solutions to $(-\\Delta )^s u+\\lambda u=u^p$ in ${B}$ with $u=0$ on $\\mathbb{R}^N\\setminus B$, are unique and nondegenerate for any $\\lambda> -\\lambda_1(B)$. This extends the very recent results in \\cite{DIS-1,Azahara-Parini} which provide uniqueness and nondegeneracy of least energy solutions.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-10-16T16:53:37.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "uniqueness",
      "nondegeneracy",
      "positive solutions",
      "first dirichlet eigenvalue",
      "nondegenerate"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}