Abstract
We construct solutions for the fractional Yamabe problem that are singular at a prescribed number of isolated points. This seems to be the first time that a gluing method is successfully applied to a non-local problem in order to construct singular solutions. There are two main steps in the proof: to construct an approximate solution by gluing half bubble towers at each singular point, and then an infinite-dimensional Lyapunov–Schmidt reduction method, that reduces the problem to an (infinite-dimensional) Toda-type system. The main technical part is the estimate of the interactions between different bubbles in the bubble towers.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11801421
Award Identifier / Grant number: 11631011
Funding source: National Science Foundation
Award Identifier / Grant number: DMS-1440140
Funding source: Ministerio de Economía y Competitividad
Award Identifier / Grant number: MTM2014-52402-C3-1-P
Award Identifier / Grant number: MTM2017-85757-P
Funding statement: Weiwei Ao was supported by NSFC (no. 11801421 and no. 11631011). Azahara DelaTorre is supported by MINECO grants MTM2014-52402-C3-1-P and MTM2017-85757-P, and the FPI-2012 fellowship, and is part of the Catalan research group 2014SGR1083. María del Mar González is supported by MINECO grants MTM2014-52402-C3-1-P and MTM2017-85757-P, the Fundación BBVA grant for Investigadores y Creadores Culturales 2016, and is part of the Barcelona Graduate School of Math and the Catalan research group 2014SGR1083. She also would like to acknowledge the NSF grant DMS-1440140 while she was in residence at the Mathematical Sciences Research Institute in Berkeley, CA, during Spring 2016. Juncheng Wei is partially supported by NSERC of Canada.
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