Frank, Rupert L. and Lenzmann, Enno and Silvestre, Luis.
(2013)
* Uniqueness of Radial Solutions for the Fractional Laplacian.*
Preprints Fachbereich Mathematik, 2013 (06).

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## Abstract

We prove general uniqueness results for radial solutions of linear and nonlinear equations involving the fractional Laplacian $(−Δ)^s$ with s∈(0,1) for any space dimensions N≥1. By extending a monotonicity formula found à la Cabré and Sire [9], we show that the linear equation

\[

(−Δ)^su+Vu=0 in \mathbb{R}^N

\]

has at most one radial and bounded solution vanishing at infinity, provided that the potential V is a radial and non-decreasing. In particular, this result implies that all radial eigenvalues of the corresponding fractional Schrödinger operator $H=(−Δ)^s+V$ are simple. Furthermore, by combining these findings on linear equations with topological bounds for a related problem on the upper half-space $\mathbb{R}^{N+1}_+$, we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation

\[

(−Δ)^sQ+Q−|Q|^αQ=0 in \mathbb{R}^N

\]

for arbitrary space dimensions N≥1 and all admissible exponents α>0. This generalizes the nondegeneracy and uniqueness result for dimension N=1 recently obtained by the first two authors in [19] and, in particular, the uniqueness result for solitary waves of the Benjamin-Ono equation found by Amick and Toland [4].

\[

(−Δ)^su+Vu=0 in \mathbb{R}^N

\]

has at most one radial and bounded solution vanishing at infinity, provided that the potential V is a radial and non-decreasing. In particular, this result implies that all radial eigenvalues of the corresponding fractional Schrödinger operator $H=(−Δ)^s+V$ are simple. Furthermore, by combining these findings on linear equations with topological bounds for a related problem on the upper half-space $\mathbb{R}^{N+1}_+$, we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation

\[

(−Δ)^sQ+Q−|Q|^αQ=0 in \mathbb{R}^N

\]

for arbitrary space dimensions N≥1 and all admissible exponents α>0. This generalizes the nondegeneracy and uniqueness result for dimension N=1 recently obtained by the first two authors in [19] and, in particular, the uniqueness result for solitary waves of the Benjamin-Ono equation found by Amick and Toland [4].

Faculties and Departments: | 05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Lenzmann) 12 Special Collections > Preprints Fachbereich Mathematik |
---|---|

UniBasel Contributors: | Lenzmann, Enno |

Item Type: | Preprint |

Publisher: | Universität Basel |

Language: | English |

Last Modified: | 12 May 2019 23:27 |

Deposited On: | 28 Mar 2019 09:52 |

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