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 ${(-\Delta )}^{s}u=f\left(u\right)$ 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 $C^{\beta }$ and when $2s+\beta <1$, the solution is not always $C^{2s+\beta -ϵ}$ for all $ϵ>0$. Instead, in general the solution $u$ is at most $C^{2s/(1-\beta )}.$