*Preprint*

**Inserted:** 25 jul 2018

**Last Updated:** 25 jul 2018

**Year:** 2018

**Abstract:**

We consider a pseudo-differential equation driven by the fractional $p$-Laplacian with $s\in(0,1)$ and $p\ge 2$ (degenerate case), with a bounded reaction $f$ and Dirichlet type conditions in a smooth domain $\Omega$. By means of barriers, a nonlocal superposition principle, and the comparison principle, we prove that any weak solution $u$ of such equation exhibits a weighted H\"older regularity up to the boundary, that is, $u/d^s\in C^\alpha(\overline\Omega)$ for some $\alpha\in(0,1)$, $d$ being the distance from the boundary.

**Keywords:**
boundary regularity, Fractional $p$-Laplacian, Weighted H\"older regularity

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