Explicit counterexamples show the Hölder exponent 2s/(1-β) is optimal for semilinear fractional Laplacian equations in one dimension when 2s≤1-β, with 2s+β optimal otherwise.
Shahgholian, Regularity issues for semilinear PDE-s (a narrative approa ch), Algebra i Analiz 27 (2015), no
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Examples of optimal H\"older regularity in semilinear equations involving the fractional Laplacian
Explicit counterexamples show the Hölder exponent 2s/(1-β) is optimal for semilinear fractional Laplacian equations in one dimension when 2s≤1-β, with 2s+β optimal otherwise.