The nonlocal mean-field equation on an interval

Abstract

We consider the fractional mean-field equation on the interval I = (−1, 1)subject to Dirichlet boundary conditions, and prove that existence holds if and only if ρ < 2π. This requires the study of blowing-up sequences of solutions. We provide a series of tools in particular which can be used (and extended) to higher-order mean field equations of non-local type.

ODE Methods in Non-Local Equations

“…Pohožaev identities for the fractional Laplacian have been considered in [24,28,47], for instance. Based on our study of ODEs with fractional Laplacian, here we derive some (new) Pohožaev identities for w a radially symmetric solution of P γ w−κw = w n+2γ n−2γ , w = w(t).…”

ODE-methods in non-local equations

“…Pohožaev identities for the fractional Laplacian have been considered in [24,39,21], for instance. Based on our study of ODEs with fractional Laplacian, here we derive some (new) Pohožaev identities for w a radially symmetric solution of (6.1) P γ w − κw = w n+2γ n−2γ , w = w(t).…”