Examples of optimal Hölder regularity in semilinear equations involving the fractional Laplacian

Abstract

We discuss the Hölder regularity of solutions to the semilinear equation involving the fractional Laplacian (−Δ)su=f(u) in one dimension. We put in evidence a new regularity phenomenon which is a combined effect of the nonlocality and the semilinearity of the equation, since it does not happen neither for local semilinear equations, nor for nonlocal linear equations. Namely, for nonlinearities f in Cβ and when 2s+β<1, the solution is not always C2s+β−ϵ for all ϵ>0. Instead, in general the solution u is at most C2s/(1−β).