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 - <p style="text-align: justify;">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.</p>