Volume 53, Issue 4
ODE Methods in Non-Local Equations

Weiwei Ao, Hardy Chan, Azahara DelaTorre, Marco A. Fontelos, María del Mar González & Juncheng Wei

J. Math. Study, 53 (2020), pp. 370-401.

Published online: 2020-12

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  • Abstract

Non-local equations cannot be treated using classical ODE theorems. Nevertheless, several new methods have been introduced in the non-local gluing scheme of our previous article; we survey and improve those, and present new applications as well. First, from the explicit symbol of the conformal fractional Laplacian, a variation of constants formula is obtained for fractional Hardy operators. We thus develop, in addition to a suitable extension in the spirit of Caffarelli–Silvestre, an equivalent formulation as an infinite system of second order constant coefficient ODEs. Classical ODE quantities like the Hamiltonian and Wrońskian may then be utilized. As applications, we obtain a Frobenius theorem and establish new Pohožaev identities. We also give a detailed proof for the non-degeneracy of the fast-decay singular solution of the fractional Lane–Emden equation.

  • AMS Subject Headings

Primary 35J61, Secondary 35R11, 53A30.

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

wwao@whu.edu.cn (Weiwei Ao)

hardy.chan@math.ethz.ch (Hardy Chan)

adelatorre@ugr.es (Azahara DelaTorre)

marco.fontelos@icmat.es (Marco A. Fontelos)

mariamar.gonzalezn@uam.es (María del Mar González)

jcwei@math.ubc.ca (Juncheng Wei)

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@Article{JMS-53-370, author = {Ao , WeiweiChan , HardyDelaTorre , AzaharaFontelos , Marco A.Mar González , María del and Wei , Juncheng}, title = {ODE Methods in Non-Local Equations}, journal = {Journal of Mathematical Study}, year = {2020}, volume = {53}, number = {4}, pages = {370--401}, abstract = {

Non-local equations cannot be treated using classical ODE theorems. Nevertheless, several new methods have been introduced in the non-local gluing scheme of our previous article; we survey and improve those, and present new applications as well. First, from the explicit symbol of the conformal fractional Laplacian, a variation of constants formula is obtained for fractional Hardy operators. We thus develop, in addition to a suitable extension in the spirit of Caffarelli–Silvestre, an equivalent formulation as an infinite system of second order constant coefficient ODEs. Classical ODE quantities like the Hamiltonian and Wrońskian may then be utilized. As applications, we obtain a Frobenius theorem and establish new Pohožaev identities. We also give a detailed proof for the non-degeneracy of the fast-decay singular solution of the fractional Lane–Emden equation.

}, issn = {2617-8702}, doi = {https://doi.org/10.4208/jms.v53n4.20.01}, url = {http://global-sci.org/intro/article_detail/jms/18507.html} }
TY - JOUR T1 - ODE Methods in Non-Local Equations AU - Ao , Weiwei AU - Chan , Hardy AU - DelaTorre , Azahara AU - Fontelos , Marco A. AU - Mar González , María del AU - Wei , Juncheng JO - Journal of Mathematical Study VL - 4 SP - 370 EP - 401 PY - 2020 DA - 2020/12 SN - 53 DO - http://doi.org/10.4208/jms.v53n4.20.01 UR - https://global-sci.org/intro/article_detail/jms/18507.html KW - ODE methods, non-local equations, fractional Hardy operators, Frobenius theorem. AB -

Non-local equations cannot be treated using classical ODE theorems. Nevertheless, several new methods have been introduced in the non-local gluing scheme of our previous article; we survey and improve those, and present new applications as well. First, from the explicit symbol of the conformal fractional Laplacian, a variation of constants formula is obtained for fractional Hardy operators. We thus develop, in addition to a suitable extension in the spirit of Caffarelli–Silvestre, an equivalent formulation as an infinite system of second order constant coefficient ODEs. Classical ODE quantities like the Hamiltonian and Wrońskian may then be utilized. As applications, we obtain a Frobenius theorem and establish new Pohožaev identities. We also give a detailed proof for the non-degeneracy of the fast-decay singular solution of the fractional Lane–Emden equation.

Weiwei Ao, Hardy Chan, Azahara DelaTorre, Marco A. Fontelos, María del Mar González & Juncheng Wei. (2020). ODE Methods in Non-Local Equations. Journal of Mathematical Study. 53 (4). 370-401. doi:10.4208/jms.v53n4.20.01
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