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Delaunay-type singular solutions for the fractional Yamabe problem

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Abstract

We construct Delaunay-type solutions for the fractional Yamabe problem with an isolated singularity

$$\begin{aligned} (-\Delta )^\gamma w= c_{n, {\gamma }}w^{\frac{n+2\gamma }{n-2\gamma }}, w>0 \quad \text{ in } \quad \mathbb R^n \backslash \{0\}. \end{aligned}$$

We follow a variational approach, in which the key is the computation of the fractional Laplacian in polar coordinates.

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Acknowledgments

The authors would like to thank the anonymous referee for his/her many suggestions and improvements of the paper.

Author information

Authors and Affiliations

  1. Universitat Politècnica de Catalunya, ETSEIB-Matematiques, Av. Diagonal 647, 08028, Barcelona, Spain

    Azahara DelaTorre & María del Mar González

  2. Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático (UMI 2807 CNRS), Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile

    Manuel del Pino

  3. Department of Mathematics, University of British Columbia, Vancouver, BC, V6T1Z2, Canada

    Juncheng Wei

Authors
  1. Azahara DelaTorre
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  2. Manuel del Pino
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  3. María del Mar González
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  4. Juncheng Wei
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Corresponding author

Correspondence to María del Mar González.

Additional information

Communicated by F. C. Marques.

A. DelaTorre is Supported by MINECO Grants MTM2011-27739-C04-01, MTM2014-52402-C3-1-P and FPI-2012 fellowship, and is part of the Catalan research group 2014SGR1083. M.d.M. González is supported by MINECO Grants MTM2011-27739-C04-01 and MTM2014-52402-C3-1-P, and is part of the Barcelona Graduate School of Math and the Catalan research group 2014SGR1083. M. del Pino has been supported by Grants Fondecyt 1150066, Fondo Basal CMM and by Nucleo Mienio CAPDE, NC130017. J. Wei is partially supported by NSERC of Canada.

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DelaTorre, A., del Pino, M., González, M.d.M. et al. Delaunay-type singular solutions for the fractional Yamabe problem. Math. Ann. 369, 597–626 (2017). https://doi.org/10.1007/s00208-016-1483-1

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  • DOI: https://doi.org/10.1007/s00208-016-1483-1

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