ODE Methods in Non-Local Equations (Journal of Mathematical Study, 2020)
Non-local equations cannot be treated using classical ODE theorems. Nevertheless, several new methods have been introduced in the non-local gluing schemeof our previous article; we survey and improve those, and present new applications aswell. First, from the explicit symbol of the conformal fractional Laplacian, a variationof constants formula is obtained for fractional Hardy operators. We thus develop, inaddition to a suitable extension in the spirit of Caffarelli–Silvestre, an equivalent formulation as an infinite system of second order constant coefficient ODEs. ClassicalODE 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 thefractional Lane–Emden equation.