Examples of optimal H\"older regularity in semilinear equations involving the fractional Laplacian

Abstract

We discuss the H\"older regularity of solutions to the semilinear equation involving the fractional Laplacian $(-\Delta)^s u=f(u)$ 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 $C^\beta$ and when $2s+\beta <1$, the solution is not always $C^{2s+\beta-\epsilon}$ for all $\epsilon >0$. Instead, in general the solution $u$ is at most $C^{2s/(1-\beta)}.$