Uniqueness of Radial Solutions for the Fractional Laplacian |
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| Authors: | Rupert L. Frank Enno Lenzmann Luis Silvestre |
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| Affiliation: | 1. Caltech Mathematics, Pasadena, CA, USA;2. University of Basel, Department of Mathematics and Computer Science, Basel, Switzerland;3. University of Chicago, Mathematics Department, Chicago, IL, USA |
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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 by Cabré and Sire , we show that the linear equation ![urn:x-wiley:00103640:media:cpa21591:cpa21591-math-0001]( "urn:x-wiley:00103640:media:cpa21591:cpa21591-math-0001") has at most one radial and bounded solution vanishing at infinity, provided that the potential V is radial and nondecreasing. 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  , we show uniqueness and nondegeneracy of ground state solutions for the nonlinear equation ![urn:x-wiley:00103640:media:cpa21591:cpa21591-math-0003]( "urn:x-wiley:00103640:media:cpa21591:cpa21591-math-0003") 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 and, in particular, the uniqueness result for solitary waves of the Benjamin‐Ono equation found by Amick and Toland .© 2016 Wiley Periodicals, Inc. |
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