Examples of optimal Hölder regularity in semilinear equations involving the fractional Laplacian
Gyula Csató, Albert Mas Blesa · Nonlinear Analysis · 2025
We discuss the Hölder regularity of solutions to the semilinear equation involving the fractional Laplacian ( − Δ ) 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 β and when 2 s + β 0 . Instead, in general the solution u is at most C 2 s / ( 1 − β ) .