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arXiv · 1910.14512

ODE-methods in non-local equations

Weiwei Ao·Hardy Chan·Azahara DelaTorre·Marco A. Fontelos·María Del Mar González·Juncheng Wei
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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 "On higher dimensional singularities for the fractional Yamabe problem: a non-local Mazzeo-Pacard program"; 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 Wro\'nskian may then be utilized. As applications, we obtain a Frobenius theorem and establish new Poho\vzaev identities. We also give a detailed proof for the non-degeneracy of the fast-decay singular solution of the fractional Lane-Emden equation.

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Weiwei Ao, Hardy Chan, Azahara DelaTorre, Marco A. Fontelos, María Del Mar González, Juncheng Wei. 2019-10-31. ODE-methods in non-local equations. https://arxiv.org/abs/1910.14512

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